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<span class="vector-toc-numb">1.1</span>Standard normal distribution</div>
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<span class="vector-toc-numb">1.2</span>General normal distribution</div>
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<span class="vector-toc-numb">1.4</span>Alternative parameterizations</div>
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<span class="vector-toc-numb">1.5</span>Cumulative distribution function</div>
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class="vector-toc-list-item vector-toc-level-2">
<a class="vector-toc-link" href="#Error_Function">
<div class="vector-toc-text">
<span class="vector-toc-numb">1.6</span>Error Function</div>
</a>
<ul id="toc-Error_Function-sublist" class="vector-toc-list">
<li id="toc-Recursive_computation_with_Taylor_series_expansion"
class="vector-toc-list-item vector-toc-level-3">
<a class="vector-toc-link" href="#Recursive_computation_with_Taylor_series_expansion">
<div class="vector-toc-text">
<span class="vector-toc-numb">1.6.1</span>Recursive computation with Taylor series expansion</div>
</a>
<ul id="toc-Recursive_computation_with_Taylor_series_expansion-sublist" class="vector-toc-list">
</ul>
</li>
<li id="toc-Using_the_Taylor_series_and_Newton&#039;s_method_for_the_inverse_function"
class="vector-toc-list-item vector-toc-level-3">
<a class="vector-toc-link" href="#Using_the_Taylor_series_and_Newton&#039;s_method_for_the_inverse_function">
<div class="vector-toc-text">
<span class="vector-toc-numb">1.6.2</span>Using the Taylor series and Newton's method for the inverse function</div>
</a>
<ul id="toc-Using_the_Taylor_series_and_Newton&#039;s_method_for_the_inverse_function-sublist" class="vector-toc-list">
</ul>
</li>
<li id="toc-Standard_deviation_and_coverage"
class="vector-toc-list-item vector-toc-level-3">
<a class="vector-toc-link" href="#Standard_deviation_and_coverage">
<div class="vector-toc-text">
<span class="vector-toc-numb">1.6.3</span>Standard deviation and coverage</div>
</a>
<ul id="toc-Standard_deviation_and_coverage-sublist" class="vector-toc-list">
</ul>
</li>
<li id="toc-Quantile_function"
class="vector-toc-list-item vector-toc-level-3">
<a class="vector-toc-link" href="#Quantile_function">
<div class="vector-toc-text">
<span class="vector-toc-numb">1.6.4</span>Quantile function</div>
</a>
<ul id="toc-Quantile_function-sublist" class="vector-toc-list">
</ul>
</li>
</ul>
</li>
</ul>
</li>
<li id="toc-Properties"
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<a class="vector-toc-link" href="#Properties">
<div class="vector-toc-text">
<span class="vector-toc-numb">2</span>Properties</div>
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<li id="toc-Symmetries_and_derivatives"
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<a class="vector-toc-link" href="#Symmetries_and_derivatives">
<div class="vector-toc-text">
<span class="vector-toc-numb">2.1</span>Symmetries and derivatives</div>
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<ul id="toc-Symmetries_and_derivatives-sublist" class="vector-toc-list">
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<li id="toc-Moments"
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<a class="vector-toc-link" href="#Moments">
<div class="vector-toc-text">
<span class="vector-toc-numb">2.2</span>Moments</div>
</a>
<ul id="toc-Moments-sublist" class="vector-toc-list">
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<li id="toc-Fourier_transform_and_characteristic_function"
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<a class="vector-toc-link" href="#Fourier_transform_and_characteristic_function">
<div class="vector-toc-text">
<span class="vector-toc-numb">2.3</span>Fourier transform and characteristic function</div>
</a>
<ul id="toc-Fourier_transform_and_characteristic_function-sublist" class="vector-toc-list">
</ul>
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<li id="toc-Moment-_and_cumulant-generating_functions"
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<a class="vector-toc-link" href="#Moment-_and_cumulant-generating_functions">
<div class="vector-toc-text">
<span class="vector-toc-numb">2.4</span>Moment- and cumulant-generating functions</div>
</a>
<ul id="toc-Moment-_and_cumulant-generating_functions-sublist" class="vector-toc-list">
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<li id="toc-Stein_operator_and_class"
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<a class="vector-toc-link" href="#Stein_operator_and_class">
<div class="vector-toc-text">
<span class="vector-toc-numb">2.5</span>Stein operator and class</div>
</a>
<ul id="toc-Stein_operator_and_class-sublist" class="vector-toc-list">
</ul>
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<li id="toc-Zero-variance_limit"
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<a class="vector-toc-link" href="#Zero-variance_limit">
<div class="vector-toc-text">
<span class="vector-toc-numb">2.6</span>Zero-variance limit</div>
</a>
<ul id="toc-Zero-variance_limit-sublist" class="vector-toc-list">
</ul>
</li>
<li id="toc-Maximum_entropy"
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<a class="vector-toc-link" href="#Maximum_entropy">
<div class="vector-toc-text">
<span class="vector-toc-numb">2.7</span>Maximum entropy</div>
</a>
<ul id="toc-Maximum_entropy-sublist" class="vector-toc-list">
</ul>
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<li id="toc-Other_properties"
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<a class="vector-toc-link" href="#Other_properties">
<div class="vector-toc-text">
<span class="vector-toc-numb">2.8</span>Other properties</div>
</a>
<ul id="toc-Other_properties-sublist" class="vector-toc-list">
</ul>
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</ul>
</li>
<li id="toc-Related_distributions"
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<a class="vector-toc-link" href="#Related_distributions">
<div class="vector-toc-text">
<span class="vector-toc-numb">3</span>Related distributions</div>
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<span>Toggle Related distributions subsection</span>
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<li id="toc-Central_limit_theorem"
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<a class="vector-toc-link" href="#Central_limit_theorem">
<div class="vector-toc-text">
<span class="vector-toc-numb">3.1</span>Central limit theorem</div>
</a>
<ul id="toc-Central_limit_theorem-sublist" class="vector-toc-list">
</ul>
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<li id="toc-Operations_and_functions_of_normal_variables"
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<a class="vector-toc-link" href="#Operations_and_functions_of_normal_variables">
<div class="vector-toc-text">
<span class="vector-toc-numb">3.2</span>Operations and functions of normal variables</div>
</a>
<ul id="toc-Operations_and_functions_of_normal_variables-sublist" class="vector-toc-list">
<li id="toc-Operations_on_a_single_normal_variable"
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<a class="vector-toc-link" href="#Operations_on_a_single_normal_variable">
<div class="vector-toc-text">
<span class="vector-toc-numb">3.2.1</span>Operations on a single normal variable</div>
</a>
<ul id="toc-Operations_on_a_single_normal_variable-sublist" class="vector-toc-list">
<li id="toc-Operations_on_two_independent_normal_variables"
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<a class="vector-toc-link" href="#Operations_on_two_independent_normal_variables">
<div class="vector-toc-text">
<span class="vector-toc-numb">3.2.1.1</span>Operations on two independent normal variables</div>
</a>
<ul id="toc-Operations_on_two_independent_normal_variables-sublist" class="vector-toc-list">
</ul>
</li>
<li id="toc-Operations_on_two_independent_standard_normal_variables"
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<a class="vector-toc-link" href="#Operations_on_two_independent_standard_normal_variables">
<div class="vector-toc-text">
<span class="vector-toc-numb">3.2.1.2</span>Operations on two independent standard normal variables</div>
</a>
<ul id="toc-Operations_on_two_independent_standard_normal_variables-sublist" class="vector-toc-list">
</ul>
</li>
</ul>
</li>
<li id="toc-Operations_on_multiple_independent_normal_variables"
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<a class="vector-toc-link" href="#Operations_on_multiple_independent_normal_variables">
<div class="vector-toc-text">
<span class="vector-toc-numb">3.2.2</span>Operations on multiple independent normal variables</div>
</a>
<ul id="toc-Operations_on_multiple_independent_normal_variables-sublist" class="vector-toc-list">
</ul>
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<li id="toc-Operations_on_multiple_correlated_normal_variables"
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<a class="vector-toc-link" href="#Operations_on_multiple_correlated_normal_variables">
<div class="vector-toc-text">
<span class="vector-toc-numb">3.2.3</span>Operations on multiple correlated normal variables</div>
</a>
<ul id="toc-Operations_on_multiple_correlated_normal_variables-sublist" class="vector-toc-list">
</ul>
</li>
</ul>
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<li id="toc-Operations_on_the_density_function"
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<a class="vector-toc-link" href="#Operations_on_the_density_function">
<div class="vector-toc-text">
<span class="vector-toc-numb">3.3</span>Operations on the density function</div>
</a>
<ul id="toc-Operations_on_the_density_function-sublist" class="vector-toc-list">
</ul>
</li>
<li id="toc-Infinite_divisibility_and_Cramér&#039;s_theorem"
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<a class="vector-toc-link" href="#Infinite_divisibility_and_Cramér&#039;s_theorem">
<div class="vector-toc-text">
<span class="vector-toc-numb">3.4</span>Infinite divisibility and Cramér's theorem</div>
</a>
<ul id="toc-Infinite_divisibility_and_Cramér&#039;s_theorem-sublist" class="vector-toc-list">
</ul>
</li>
<li id="toc-Bernstein&#039;s_theorem"
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<a class="vector-toc-link" href="#Bernstein&#039;s_theorem">
<div class="vector-toc-text">
<span class="vector-toc-numb">3.5</span>Bernstein's theorem</div>
</a>
<ul id="toc-Bernstein&#039;s_theorem-sublist" class="vector-toc-list">
</ul>
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<li id="toc-Extensions"
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<a class="vector-toc-link" href="#Extensions">
<div class="vector-toc-text">
<span class="vector-toc-numb">3.6</span>Extensions</div>
</a>
<ul id="toc-Extensions-sublist" class="vector-toc-list">
</ul>
</li>
</ul>
</li>
<li id="toc-Statistical_inference"
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<a class="vector-toc-link" href="#Statistical_inference">
<div class="vector-toc-text">
<span class="vector-toc-numb">4</span>Statistical inference</div>
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<li id="toc-Estimation_of_parameters"
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<a class="vector-toc-link" href="#Estimation_of_parameters">
<div class="vector-toc-text">
<span class="vector-toc-numb">4.1</span>Estimation of parameters</div>
</a>
<ul id="toc-Estimation_of_parameters-sublist" class="vector-toc-list">
<li id="toc-Sample_mean"
class="vector-toc-list-item vector-toc-level-3">
<a class="vector-toc-link" href="#Sample_mean">
<div class="vector-toc-text">
<span class="vector-toc-numb">4.1.1</span>Sample mean</div>
</a>
<ul id="toc-Sample_mean-sublist" class="vector-toc-list">
</ul>
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<li id="toc-Sample_variance"
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<a class="vector-toc-link" href="#Sample_variance">
<div class="vector-toc-text">
<span class="vector-toc-numb">4.1.2</span>Sample variance</div>
</a>
<ul id="toc-Sample_variance-sublist" class="vector-toc-list">
</ul>
</li>
</ul>
</li>
<li id="toc-Confidence_intervals"
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<a class="vector-toc-link" href="#Confidence_intervals">
<div class="vector-toc-text">
<span class="vector-toc-numb">4.2</span>Confidence intervals</div>
</a>
<ul id="toc-Confidence_intervals-sublist" class="vector-toc-list">
</ul>
</li>
<li id="toc-Normality_tests"
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<a class="vector-toc-link" href="#Normality_tests">
<div class="vector-toc-text">
<span class="vector-toc-numb">4.3</span>Normality tests</div>
</a>
<ul id="toc-Normality_tests-sublist" class="vector-toc-list">
</ul>
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<li id="toc-Bayesian_analysis_of_the_normal_distribution"
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<a class="vector-toc-link" href="#Bayesian_analysis_of_the_normal_distribution">
<div class="vector-toc-text">
<span class="vector-toc-numb">4.4</span>Bayesian analysis of the normal distribution</div>
</a>
<ul id="toc-Bayesian_analysis_of_the_normal_distribution-sublist" class="vector-toc-list">
<li id="toc-Sum_of_two_quadratics"
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<a class="vector-toc-link" href="#Sum_of_two_quadratics">
<div class="vector-toc-text">
<span class="vector-toc-numb">4.4.1</span>Sum of two quadratics</div>
</a>
<ul id="toc-Sum_of_two_quadratics-sublist" class="vector-toc-list">
<li id="toc-Scalar_form"
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<a class="vector-toc-link" href="#Scalar_form">
<div class="vector-toc-text">
<span class="vector-toc-numb">4.4.1.1</span>Scalar form</div>
</a>
<ul id="toc-Scalar_form-sublist" class="vector-toc-list">
</ul>
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<li id="toc-Vector_form"
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<a class="vector-toc-link" href="#Vector_form">
<div class="vector-toc-text">
<span class="vector-toc-numb">4.4.1.2</span>Vector form</div>
</a>
<ul id="toc-Vector_form-sublist" class="vector-toc-list">
</ul>
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<li id="toc-Sum_of_differences_from_the_mean"
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<a class="vector-toc-link" href="#Sum_of_differences_from_the_mean">
<div class="vector-toc-text">
<span class="vector-toc-numb">4.4.2</span>Sum of differences from the mean</div>
</a>
<ul id="toc-Sum_of_differences_from_the_mean-sublist" class="vector-toc-list">
</ul>
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</li>
<li id="toc-With_known_variance"
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<a class="vector-toc-link" href="#With_known_variance">
<div class="vector-toc-text">
<span class="vector-toc-numb">4.5</span>With known variance</div>
</a>
<ul id="toc-With_known_variance-sublist" class="vector-toc-list">
<li id="toc-With_known_mean"
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<a class="vector-toc-link" href="#With_known_mean">
<div class="vector-toc-text">
<span class="vector-toc-numb">4.5.1</span>With known mean</div>
</a>
<ul id="toc-With_known_mean-sublist" class="vector-toc-list">
</ul>
</li>
<li id="toc-With_unknown_mean_and_unknown_variance"
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<a class="vector-toc-link" href="#With_unknown_mean_and_unknown_variance">
<div class="vector-toc-text">
<span class="vector-toc-numb">4.5.2</span>With unknown mean and unknown variance</div>
</a>
<ul id="toc-With_unknown_mean_and_unknown_variance-sublist" class="vector-toc-list">
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<li id="toc-Occurrence_and_applications"
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<a class="vector-toc-link" href="#Occurrence_and_applications">
<div class="vector-toc-text">
<span class="vector-toc-numb">5</span>Occurrence and applications</div>
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<li id="toc-Exact_normality"
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<a class="vector-toc-link" href="#Exact_normality">
<div class="vector-toc-text">
<span class="vector-toc-numb">5.1</span>Exact normality</div>
</a>
<ul id="toc-Exact_normality-sublist" class="vector-toc-list">
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<a class="vector-toc-link" href="#Approximate_normality">
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<span class="vector-toc-numb">5.2</span>Approximate normality</div>
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<li id="toc-Assumed_normality"
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<a class="vector-toc-link" href="#Assumed_normality">
<div class="vector-toc-text">
<span class="vector-toc-numb">5.3</span>Assumed normality</div>
</a>
<ul id="toc-Assumed_normality-sublist" class="vector-toc-list">
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<li id="toc-Methodological_problems_and_peer_review"
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<a class="vector-toc-link" href="#Methodological_problems_and_peer_review">
<div class="vector-toc-text">
<span class="vector-toc-numb">5.4</span>Methodological problems and peer review</div>
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<span class="vector-toc-numb">6</span>Computational methods</div>
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<li id="toc-Generating_values_from_normal_distribution"
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<a class="vector-toc-link" href="#Generating_values_from_normal_distribution">
<div class="vector-toc-text">
<span class="vector-toc-numb">6.1</span>Generating values from normal distribution</div>
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<ul id="toc-Generating_values_from_normal_distribution-sublist" class="vector-toc-list">
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<li id="toc-Numerical_approximations_for_the_normal_cumulative_distribution_function_and_normal_quantile_function"
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<div class="vector-toc-text">
<span class="vector-toc-numb">6.2</span>Numerical approximations for the normal cumulative distribution function and normal quantile function</div>
</a>
<ul id="toc-Numerical_approximations_for_the_normal_cumulative_distribution_function_and_normal_quantile_function-sublist" class="vector-toc-list">
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<li id="toc-History"
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<a class="vector-toc-link" href="#History">
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<span class="vector-toc-numb">7</span>History</div>
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<span>Toggle History subsection</span>
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<a class="vector-toc-link" href="#Development">
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<span class="vector-toc-numb">7.1</span>Development</div>
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<ul id="toc-Development-sublist" class="vector-toc-list">
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<li id="toc-Naming"
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<a class="vector-toc-link" href="#Naming">
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<span class="vector-toc-numb">7.2</span>Naming</div>
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<ul id="toc-Naming-sublist" class="vector-toc-list">
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<li id="toc-See_also"
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<a class="vector-toc-link" href="#See_also">
<div class="vector-toc-text">
<span class="vector-toc-numb">8</span>See also</div>
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<ul id="toc-See_also-sublist" class="vector-toc-list">
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<li id="toc-Notes"
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<a class="vector-toc-link" href="#Notes">
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<span class="vector-toc-numb">9</span>Notes</div>
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<ul id="toc-Notes-sublist" class="vector-toc-list">
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<li id="toc-References"
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<a class="vector-toc-link" href="#References">
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<span class="vector-toc-numb">10</span>References</div>
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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Normal distribution</span></h1>
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<li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Normalverteilung" title="Normalverteilung Alemannic" lang="gsw" hreflang="gsw" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D9%88%D8%B2%D9%8A%D8%B9_%D8%A7%D8%AD%D8%AA%D9%85%D8%A7%D9%84%D9%8A_%D8%B7%D8%A8%D9%8A%D8%B9%D9%8A" title="توزيع احتمالي طبيعي Arabic" lang="ar" hreflang="ar" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Distribuci%C3%B3n_normal" title="Distribución normal Asturian" lang="ast" hreflang="ast" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Normal_paylanma" title="Normal paylanma Azerbaijani" lang="az" hreflang="az" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D9%86%D9%88%D8%B1%D9%85%D8%A7%D9%84_%D8%AF%D8%A7%D8%BA%DB%8C%D9%84%DB%8C%D9%85" title="نورمال داغیلیم South Azerbaijani" lang="azb" hreflang="azb" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Si%C3%B4ng-th%C3%A0i_hun-p%C3%B2%CD%98" title="Siông-thài hun-pò͘ Min Nan Chinese" lang="nan" hreflang="nan" class="interlanguage-link-target"><span>Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9D%D0%B0%D1%80%D0%BC%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0%D0%B5_%D1%80%D0%B0%D0%B7%D0%BC%D0%B5%D1%80%D0%BA%D0%B0%D0%B2%D0%B0%D0%BD%D0%BD%D0%B5" title="Нармальнае размеркаванне Belarusian" lang="be" hreflang="be" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9D%D0%BE%D1%80%D0%BC%D0%B0%D0%BB%D0%BD%D0%BE_%D1%80%D0%B0%D0%B7%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B5%D0%BD%D0%B8%D0%B5" title="Нормално разпределение Bulgarian" lang="bg" hreflang="bg" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Normalna_raspodjela" title="Normalna raspodjela Bosnian" lang="bs" hreflang="bs" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Distribuci%C3%B3_normal" title="Distribució normal Catalan" lang="ca" hreflang="ca" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%93%D0%B0%D1%83%D1%81%D1%81_%D0%B2%D0%B0%D0%BB%D0%B5%C3%A7%C4%95%D0%B2%C4%95" title="Гаусс валеçĕвĕ Chuvash" lang="cv" hreflang="cv" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Norm%C3%A1ln%C3%AD_rozd%C4%9Blen%C3%AD" title="Normální rozdělení Czech" lang="cs" hreflang="cs" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Dosraniad_normal" title="Dosraniad normal Welsh" lang="cy" hreflang="cy" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Normalfordeling" title="Normalfordeling Danish" lang="da" hreflang="da" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="inte
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<div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div>
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<div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Probability distribution</div>
<style data-mw-deduplicate="TemplateStyles:r1033289096">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}</style><div role="note" class="hatnote navigation-not-searchable">"Bell curve" redirects here. For other uses, see <a href="/wiki/Bell_curve_(disambiguation)" class="mw-disambig" title="Bell curve (disambiguation)">Bell curve (disambiguation)</a>.</div>
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<style data-mw-deduplicate="TemplateStyles:r1066479718">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}body.skin-minerva .mw-parser-output .infobox-header,body.skin-minerva .mw-parser-output .infobox-subheader,body.skin-minerva .mw-parser-output .infobox-above,body.skin-minerva .mw-parser-output .infobox-title,body.skin-minerva .mw-parser-output .infobox-image,body.skin-minerva .mw-parser-output .infobox-full-data,body.skin-minerva .mw-parser-output .infobox-below{text-align:center}</style><style data-mw-deduplicate="TemplateStyles:r1046248152">.mw-parser-output .ib-prob-dist{border-collapse:collapse;width:20em}.mw-parser-output .ib-prob-dist td,.mw-parser-output .ib-prob-dist th{border:1px solid #a2a9b1}.mw-parser-output .ib-prob-dist .infobox-subheader{text-align:left}.mw-parser-output .ib-prob-dist-image{background:#ddd;font-weight:bold;text-align:center}</style><table class="infobox ib-prob-dist"><caption class="infobox-title">Normal distribution</caption><tbody><tr><td colspan="4" class="infobox-image">
<div class="ib-prob-dist-image">Probability density function</div><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:Normal_Distribution_PDF.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Normal_Distribution_PDF.svg/220px-Normal_Distribution_PDF.svg.png" decoding="async" width="220" height="141" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Normal_Distribution_PDF.svg/330px-Normal_Distribution_PDF.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/74/Normal_Distribution_PDF.svg/440px-Normal_Distribution_PDF.svg.png 2x" data-file-width="720" data-file-height="460" /></a></span><div class="infobox-caption">The red curve is the <i>standard normal distribution</i></div></td></tr><tr><td colspan="4" class="infobox-image">
<div class="ib-prob-dist-image">Cumulative distribution function</div><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:Normal_Distribution_CDF.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Normal_Distribution_CDF.svg/220px-Normal_Distribution_CDF.svg.png" decoding="async" width="220" height="141" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Normal_Distribution_CDF.svg/330px-Normal_Distribution_CDF.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Normal_Distribution_CDF.svg/440px-Normal_Distribution_CDF.svg.png 2x" data-file-width="720" data-file-height="460" /></a></span></td></tr><tr><th scope="row" class="infobox-label">Notation</th><td colspan="3" class="infobox-data">
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}(\mu ,\sigma ^{2})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
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<mi class="MJX-tex-caligraphic" mathvariant="script">N</mi>
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<mo stretchy="false">(</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>,</mo>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}(\mu ,\sigma ^{2})}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/863304aaa42a945f2f07d79facc3d2eebc845ce7" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.062ex; width:8.966ex; height:3.176ex;" alt="{\mathcal {N}}(\mu ,\sigma ^{2})"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Statistical_parameter" title="Statistical parameter">Parameters</a></th><td colspan="3" class="infobox-data">
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \in \mathbb {R} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
<mo>&#x2208;<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
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<annotation encoding="application/x-tex">{\displaystyle \mu \in \mathbb {R} }</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9a48f0e84328dc53dec2ad301bb321c00dcf422" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.92ex; height:2.676ex;" alt="{\displaystyle \mu \in \mathbb {R} }"></span> = mean (<a href="/wiki/Location_parameter" title="Location parameter">location</a>)<br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}\in \mathbb {R} _{&gt;0}}">
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<msup>
<mi>&#x03C3;<!-- σ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}\in \mathbb {R} _{&gt;0}}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b81059aaa2db87802908711c7527c0e1d65fc06" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.236ex; height:3.009ex;" alt="{\displaystyle \sigma ^{2}\in \mathbb {R} _{&gt;0}}"></span> = variance (squared <a href="/wiki/Scale_parameter" title="Scale parameter">scale</a>)<br /></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Support_(mathematics)" title="Support (mathematics)">Support</a></th><td colspan="3" class="infobox-data">
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {R} }">
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<mo>&#x2208;<!-- ∈ --></mo>
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<mi mathvariant="double-struck">R</mi>
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<annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {R} }</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9c6d458566aec47a7259762034790c8981aefab" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.848ex; height:2.176ex;" alt="{\displaystyle x\in \mathbb {R} }"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Probability_density_function" title="Probability density function">PDF</a></th><td colspan="3" class="infobox-data">
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}}}">
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<mi>&#x03C3;<!-- σ --></mi>
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<mo>)</mo>
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<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}}}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a45cef4ca1e2fcd4d367ecff5806d8a2878d3821" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:16.93ex; height:6.676ex;" alt="{\displaystyle {\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">CDF</a></th><td colspan="3" class="infobox-data">
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi \left({\frac {x-\mu }{\sigma }}\right)={\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)\right]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
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<mi>&#x03C3;<!-- σ --></mi>
</mfrac>
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<mo>)</mo>
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<mo>(</mo>
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<mi>&#x03C3;<!-- σ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \Phi \left({\frac {x-\mu }{\sigma }}\right)={\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)\right]}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0fed43e25966344745178c406f04b15d0fa3783" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:36.321ex; height:6.509ex;" alt="{\displaystyle \Phi \left({\frac {x-\mu }{\sigma }}\right)={\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)\right]}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Quantile_function" title="Quantile function">Quantile</a></th><td colspan="3" class="infobox-data">
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu +\sigma {\sqrt {2}}\operatorname {erf} ^{-1}(2p-1)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
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<mn>2</mn>
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<msup>
<mi>erf</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
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</msup>
<mo>&#x2061;<!-- --></mo>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mi>p</mi>
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu +\sigma {\sqrt {2}}\operatorname {erf} ^{-1}(2p-1)}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/510cf4025ba141645f21c7f06866867648d5bc21" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.19ex; height:3.176ex;" alt="{\displaystyle \mu +\sigma {\sqrt {2}}\operatorname {erf} ^{-1}(2p-1)}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Expected_value" title="Expected value">Mean</a></th><td colspan="3" class="infobox-data">
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="\mu "></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Median" title="Median">Median</a></th><td colspan="3" class="infobox-data">
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="\mu "></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Mode_(statistics)" title="Mode (statistics)">Mode</a></th><td colspan="3" class="infobox-data">
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<mi>&#x03BC;<!-- μ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="\mu "></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Variance" title="Variance">Variance</a></th><td colspan="3" class="infobox-data">
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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<annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="\sigma ^{2}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Average_absolute_deviation" title="Average absolute deviation">MAD</a></th><td colspan="3" class="infobox-data">
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma {\sqrt {2}}\,\operatorname {erf} ^{-1}(1/2)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
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<mspace width="thinmathspace" />
<msup>
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<mrow class="MJX-TeXAtom-ORD">
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<mo>&#x2061;<!-- --></mo>
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<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle \sigma {\sqrt {2}}\,\operatorname {erf} ^{-1}(1/2)}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790fa1d2cbd04a5c6face5a0351857e0e15ed8e3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.488ex; height:3.176ex;" alt="{\displaystyle \sigma {\sqrt {2}}\,\operatorname {erf} ^{-1}(1/2)}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Skewness" title="Skewness">Skewness</a></th><td colspan="3" class="infobox-data">
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">
<semantics>
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<mstyle displaystyle="true" scriptlevel="0">
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Excess_kurtosis" class="mw-redirect" title="Excess kurtosis">Ex. kurtosis</a></th><td colspan="3" class="infobox-data">
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Information_entropy" class="mw-redirect" title="Information entropy">Entropy</a></th><td colspan="3" class="infobox-data">
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}\ln(2\pi \sigma ^{2})+{\frac {1}{2}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
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</mfrac>
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<mi>ln</mi>
<mo>&#x2061;<!-- --></mo>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mi>&#x03C0;<!-- π --></mi>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
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<mo stretchy="false">)</mo>
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<mfrac>
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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}\ln(2\pi \sigma ^{2})+{\frac {1}{2}}}</annotation>
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<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(\mu t+\sigma ^{2}t^{2}/2)}">
<semantics>
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<mstyle displaystyle="true" scriptlevel="0">
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<mo>&#x2061;<!-- --></mo>
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<mi>&#x03BC;<!-- μ --></mi>
<mi>t</mi>
<mo>+</mo>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mrow class="MJX-TeXAtom-ORD">
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<mo stretchy="false">)</mo>
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<annotation encoding="application/x-tex">{\displaystyle \exp(\mu t+\sigma ^{2}t^{2}/2)}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b73a531ccd6d15d03c0b80f4707641f57bfd521" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.047ex; height:3.176ex;" alt="{\displaystyle \exp(\mu t+\sigma ^{2}t^{2}/2)}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Characteristic_function_(probability_theory)" title="Characteristic function (probability theory)">CF</a></th><td colspan="3" class="infobox-data">
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(i\mu t-\sigma ^{2}t^{2}/2)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>exp</mi>
<mo>&#x2061;<!-- --></mo>
<mo stretchy="false">(</mo>
<mi>i</mi>
<mi>&#x03BC;<!-- μ --></mi>
<mi>t</mi>
<mo>&#x2212;<!-- --></mo>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
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</msup>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
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</msup>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
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<mo stretchy="false">)</mo>
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<annotation encoding="application/x-tex">{\displaystyle \exp(i\mu t-\sigma ^{2}t^{2}/2)}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8876b3855a069ca3b9464f56ffd8ab9c4b191c50" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.849ex; height:3.176ex;" alt="{\displaystyle \exp(i\mu t-\sigma ^{2}t^{2}/2)}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Fisher_information" title="Fisher information">Fisher information</a></th><td colspan="3" class="infobox-data">
<p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {I}}(\mu ,\sigma )={\begin{pmatrix}1/\sigma ^{2}&amp;0\\0&amp;2/\sigma ^{2}\end{pmatrix}}}">
<semantics>
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<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {I}}(\mu ,\sigma ^{2})={\begin{pmatrix}1/\sigma ^{2}&amp;0\\0&amp;1/(2\sigma ^{4})\end{pmatrix}}}">
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2047e1545c0f223d425a34cb783fb2928a87dc2e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.671ex; margin-left: -0.069ex; width:30.175ex; height:6.509ex;" alt="{\displaystyle {\mathcal {I}}(\mu ,\sigma ^{2})={\begin{pmatrix}1/\sigma ^{2}&amp;0\\0&amp;1/(2\sigma ^{4})\end{pmatrix}}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Kullback%E2%80%93Leibler_divergence" title="KullbackLeibler divergence">KullbackLeibler divergence</a></th><td colspan="3" class="infobox-data">
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {1 \over 2}\left\{\left({\frac {\sigma _{0}}{\sigma _{1}}}\right)^{2}+{\frac {(\mu _{1}-\mu _{0})^{2}}{\sigma _{1}^{2}}}-1+\ln {\sigma _{1}^{2} \over \sigma _{0}^{2}}\right\}}">
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<annotation encoding="application/x-tex">{\displaystyle {1 \over 2}\left\{\left({\frac {\sigma _{0}}{\sigma _{1}}}\right)^{2}+{\frac {(\mu _{1}-\mu _{0})^{2}}{\sigma _{1}^{2}}}-1+\ln {\sigma _{1}^{2} \over \sigma _{0}^{2}}\right\}}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd1668ab76d345b5cd802c87a9cd3c4f8e735f30" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:40.508ex; height:7.509ex;" alt="{\displaystyle {1 \over 2}\left\{\left({\frac {\sigma _{0}}{\sigma _{1}}}\right)^{2}+{\frac {(\mu _{1}-\mu _{0})^{2}}{\sigma _{1}^{2}}}-1+\ln {\sigma _{1}^{2} \over \sigma _{0}^{2}}\right\}}"></span></td></tr></tbody></table>
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<ul><li><a href="/wiki/Probability" title="Probability">Probability</a>
<ul><li><a href="/wiki/Probability_axioms" title="Probability axioms">Axioms</a></li></ul></li>
<li><a href="/wiki/Determinism" title="Determinism">Determinism</a>
<ul><li><a href="/wiki/Deterministic_system" title="Deterministic system">System</a></li></ul></li>
<li><a href="/wiki/Indeterminism" title="Indeterminism">Indeterminism</a></li>
<li><a href="/wiki/Randomness" title="Randomness">Randomness</a></li></ul></td>
</tr><tr><td class="sidebar-content">
<ul><li><a href="/wiki/Probability_space" title="Probability space">Probability space</a></li>
<li><a href="/wiki/Sample_space" title="Sample space">Sample space</a></li>
<li><a href="/wiki/Event_(probability_theory)" title="Event (probability theory)">Event</a>
<ul><li><a href="/wiki/Collectively_exhaustive_events" title="Collectively exhaustive events">Collectively exhaustive events</a></li>
<li><a href="/wiki/Elementary_event" title="Elementary event">Elementary event</a></li>
<li><a href="/wiki/Mutual_exclusivity" title="Mutual exclusivity">Mutual exclusivity</a></li>
<li><a href="/wiki/Outcome_(probability)" title="Outcome (probability)">Outcome</a></li>
<li><a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li></ul></li>
<li><a href="/wiki/Experiment_(probability_theory)" title="Experiment (probability theory)">Experiment</a>
<ul><li><a href="/wiki/Bernoulli_trial" title="Bernoulli trial">Bernoulli trial</a></li></ul></li>
<li><a href="/wiki/Probability_distribution" title="Probability distribution">Probability distribution</a>
<ul><li><a href="/wiki/Bernoulli_distribution" title="Bernoulli distribution">Bernoulli distribution</a></li>
<li><a href="/wiki/Binomial_distribution" title="Binomial distribution">Binomial distribution</a></li>
<li><a href="/wiki/Exponential_distribution" title="Exponential distribution">Exponential distribution</a></li>
<li><a class="mw-selflink selflink">Normal distribution</a></li>
<li><a href="/wiki/Pareto_distribution" title="Pareto distribution">Pareto distribution</a></li>
<li><a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson distribution</a></li></ul></li>
<li><a href="/wiki/Probability_measure" title="Probability measure">Probability measure</a></li>
<li><a href="/wiki/Random_variable" title="Random variable">Random variable</a>
<ul><li><a href="/wiki/Bernoulli_process" title="Bernoulli process">Bernoulli process</a></li>
<li><a href="/wiki/Continuous_or_discrete_variable" title="Continuous or discrete variable">Continuous or discrete</a></li>
<li><a href="/wiki/Expected_value" title="Expected value">Expected value</a></li>
<li><a href="/wiki/Markov_chain" title="Markov chain">Markov chain</a></li>
<li><a href="/wiki/Realization_(probability)" title="Realization (probability)">Observed value</a></li>
<li><a href="/wiki/Random_walk" title="Random walk">Random walk</a></li>
<li><a href="/wiki/Stochastic_process" title="Stochastic process">Stochastic process</a></li></ul></li></ul></td>
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<ul><li><a href="/wiki/Complementary_event" title="Complementary event">Complementary event</a></li>
<li><a href="/wiki/Joint_probability_distribution" title="Joint probability distribution">Joint probability</a></li>
<li><a href="/wiki/Marginal_distribution" title="Marginal distribution">Marginal probability</a></li>
<li><a href="/wiki/Conditional_probability" title="Conditional probability">Conditional probability</a></li></ul></td>
</tr><tr><td class="sidebar-content">
<ul><li><a href="/wiki/Independence_(probability_theory)" title="Independence (probability theory)">Independence</a></li>
<li><a href="/wiki/Conditional_independence" title="Conditional independence">Conditional independence</a></li>
<li><a href="/wiki/Law_of_total_probability" title="Law of total probability">Law of total probability</a></li>
<li><a href="/wiki/Law_of_large_numbers" title="Law of large numbers">Law of large numbers</a></li>
<li><a href="/wiki/Bayes%27_theorem" title="Bayes&#39; theorem">Bayes' theorem</a></li>
<li><a href="/wiki/Boole%27s_inequality" title="Boole&#39;s inequality">Boole's inequality</a></li></ul></td>
</tr><tr><td class="sidebar-content">
<ul><li><a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a></li>
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<p>In <a href="/wiki/Statistics" title="Statistics">statistics</a>, a <b>normal distribution</b> or <b>Gaussian distribution</b> is a type of <a href="/wiki/Continuous_probability_distribution" class="mw-redirect" title="Continuous probability distribution">continuous probability distribution</a> for a <a href="/wiki/Real_number" title="Real number">real-valued</a> <a href="/wiki/Random_variable" title="Random variable">random variable</a>. The general form of its <a href="/wiki/Probability_density_function" title="Probability density function">probability density function</a> is
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}}}">
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<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
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<mi>&#x03C3;<!-- σ --></mi>
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<mi>&#x03C3;<!-- σ --></mi>
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<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}}}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00cb9b2c9b866378626bcfa45c86a6de2f2b2e40" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:24.446ex; height:6.676ex;" alt="{\displaystyle f(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}}}"></span></dd></dl>
<p>The parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">
<semantics>
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<mi>&#x03BC;<!-- μ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="\mu "></span> is the <a href="/wiki/Mean" title="Mean">mean</a> or <a href="/wiki/Expected_value" title="Expected value">expectation</a> of the distribution (and also its <a href="/wiki/Median" title="Median">median</a> and <a href="/wiki/Mode_(statistics)" title="Mode (statistics)">mode</a>), while the parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03C3;<!-- σ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="\sigma "></span> is its <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviation</a>. The <a href="/wiki/Variance" title="Variance">variance</a> of the distribution is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="\sigma ^{2}"></span>. A random variable with a Gaussian distribution is said to be <b>normally distributed</b>, and is called a <b>normal deviate</b>.
</p><p>Normal distributions are important in <a href="/wiki/Statistics" title="Statistics">statistics</a> and are often used in the <a href="/wiki/Natural_science" title="Natural science">natural</a> and <a href="/wiki/Social_science" title="Social science">social sciences</a> to represent real-valued <a href="/wiki/Random_variable" title="Random variable">random variables</a> whose distributions are not known.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2">&#91;2&#93;</a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3">&#91;3&#93;</a></sup> Their importance is partly due to the <a href="/wiki/Central_limit_theorem" title="Central limit theorem">central limit theorem</a>. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution <a href="/wiki/Convergence_in_distribution" class="mw-redirect" title="Convergence in distribution">converges</a> to a normal distribution as the number of samples increases. Therefore, physical quantities that are expected to be the sum of many independent processes, such as <a href="/wiki/Measurement_error" class="mw-redirect" title="Measurement error">measurement errors</a>, often have distributions that are nearly normal.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4">&#91;4&#93;</a></sup>
</p><p>Moreover, Gaussian distributions have some unique properties that are valuable in analytic studies. For instance, any <a href="/wiki/Linear_combination" title="Linear combination">linear combination</a> of a fixed collection of independent normal deviates is a normal deviate. Many results and methods, such as <a href="/wiki/Propagation_of_uncertainty" title="Propagation of uncertainty">propagation of uncertainty</a> and <a href="/wiki/Least_squares" title="Least squares">least squares</a><sup id="cite_ref-5" class="reference"><a href="#cite_note-5">&#91;5&#93;</a></sup> parameter fitting, can be derived analytically in explicit form when the relevant variables are normally distributed.
</p><p>A normal distribution is sometimes informally called a <b>bell curve</b>.<sup id="cite_ref-mathsisfun_6-0" class="reference"><a href="#cite_note-mathsisfun-6">&#91;6&#93;</a></sup> However, many other distributions are bell-shaped (such as the <a href="/wiki/Cauchy_distribution" title="Cauchy distribution">Cauchy</a>, <a href="/wiki/Student%27s_t-distribution" title="Student&#39;s t-distribution">Student's <i>t</i></a>, and <a href="/wiki/Logistic_distribution" title="Logistic distribution">logistic</a> distributions). For other names, see <i><a href="#Naming">Naming</a></i>.
</p><p>The <a href="/wiki/Univariate_distribution" title="Univariate distribution">univariate probability distribution</a> is generalized for vectors in the <a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">multivariate normal distribution</a> and for matrices in the <a href="/wiki/Matrix_normal_distribution" title="Matrix normal distribution">matrix normal distribution</a>.
</p>
<meta property="mw:PageProp/toc" />
<h2><span class="mw-headline" id="Definitions">Definitions</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=1" title="Edit section: Definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h2>
<h3><span class="mw-headline" id="Standard_normal_distribution">Standard normal distribution</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=2" title="Edit section: Standard normal distribution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
<p>The simplest case of a normal distribution is known as the <b>standard normal distribution</b> or <b>unit normal distribution</b>. This is a special case when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu =0}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3753282c0ad2ea1e7d63f39425efd13c37da3169" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.663ex; height:2.676ex;" alt="\mu =0"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma =1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03C3;<!-- σ --></mi>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma =1}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f759e9b01b4c117d116da9f6d0e635b2247ee502" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="\sigma =1"></span>, and it is described by this <a href="/wiki/Probability_density_function" title="Probability density function">probability density function</a> (or density):
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (z)={\frac {e^{-z^{2}/2}}{\sqrt {2\pi }}}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03C6;<!-- φ --></mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
</mrow>
</msup>
<msqrt>
<mn>2</mn>
<mi>&#x03C0;<!-- π --></mi>
</msqrt>
</mfrac>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \varphi (z)={\frac {e^{-z^{2}/2}}{\sqrt {2\pi }}}.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92bc50fa87653a2d4f78372c1d2c6fe1d7f3c3d3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:14.84ex; height:7.009ex;" alt="{\displaystyle \varphi (z)={\frac {e^{-z^{2}/2}}{\sqrt {2\pi }}}.}"></span></dd></dl>
<p>The variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>z</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="z"></span> has a mean of 0 and a variance and standard deviation of 1. The density <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (z)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03C6;<!-- φ --></mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \varphi (z)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/329275ab4ea8747ba285ee26e5c1b099f8eb3a36" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="\varphi (z)"></span> has its peak <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/{\sqrt {2\pi }}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>2</mn>
<mi>&#x03C0;<!-- π --></mi>
</msqrt>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 1/{\sqrt {2\pi }}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7292157fbd787f538eab0aedd2f101441a04aff" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.755ex; height:3.176ex;" alt="1/{\sqrt {2\pi }}"></span> at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>z</mi>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z=0}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b92bfc06485cc90286474b14a516a68d8bfdd7b3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.349ex; height:2.176ex;" alt="z=0"></span> and <a href="/wiki/Inflection_point" title="Inflection point">inflection points</a> at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=+1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>z</mi>
<mo>=</mo>
<mo>+</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z=+1}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92886841f521351bdfff981c21c039896c283127" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.157ex; height:2.343ex;" alt="{\displaystyle z=+1}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=-1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>z</mi>
<mo>=</mo>
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z=-1}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af5d92c041c1ddaa688b8f7f68d333e107e93709" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.157ex; height:2.343ex;" alt="z=-1"></span>.
</p><p>Although the density above is most commonly known as the <i>standard normal,</i> a few authors have used that term to describe other versions of the normal distribution. <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a>, for example, once defined the standard normal as
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (z)={\frac {e^{-z^{2}}}{\sqrt {\pi }}},}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03C6;<!-- φ --></mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</msup>
<msqrt>
<mi>&#x03C0;<!-- π --></mi>
</msqrt>
</mfrac>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \varphi (z)={\frac {e^{-z^{2}}}{\sqrt {\pi }}},}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d08fa7405690b759b8c863dfd86313b7bfe1471" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:13.196ex; height:7.009ex;" alt="{\displaystyle \varphi (z)={\frac {e^{-z^{2}}}{\sqrt {\pi }}},}"></span></dd></dl>
<p>which has a variance of 1/2, and <a href="/wiki/Stephen_Stigler" title="Stephen Stigler">Stephen Stigler</a><sup id="cite_ref-7" class="reference"><a href="#cite_note-7">&#91;7&#93;</a></sup> once defined the standard normal as
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (z)=e^{-\pi z^{2}},}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03C6;<!-- φ --></mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<mi>&#x03C0;<!-- π --></mi>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</msup>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \varphi (z)=e^{-\pi z^{2}},}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67d6310ec49a504c34d6800009a9467b8f5302e2" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.302ex; height:3.509ex;" alt="{\displaystyle \varphi (z)=e^{-\pi z^{2}},}"></span></dd></dl>
<p>which has a simple functional form and a variance of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}=1/(2\pi ).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>=</mo>
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mi>&#x03C0;<!-- π --></mi>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}=1/(2\pi ).}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce8ce785bd88de7656a4a229c283ce6fdf8c72aa" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.759ex; height:3.176ex;" alt="{\displaystyle \sigma ^{2}=1/(2\pi ).}"></span>
</p>
<h3><span class="mw-headline" id="General_normal_distribution">General normal distribution</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=3" title="Edit section: General normal distribution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
<p>Every normal distribution is a version of the standard normal distribution, whose domain has been stretched by a factor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03C3;<!-- σ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="\sigma "></span> (the standard deviation) and then translated by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="\mu "></span> (the mean value):
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x\mid \mu ,\sigma ^{2})={\frac {1}{\sigma }}\varphi \left({\frac {x-\mu }{\sigma }}\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>&#x2223;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>,</mo>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mi>&#x03C3;<!-- σ --></mi>
</mfrac>
</mrow>
<mi>&#x03C6;<!-- φ --></mi>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>x</mi>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
</mrow>
<mi>&#x03C3;<!-- σ --></mi>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(x\mid \mu ,\sigma ^{2})={\frac {1}{\sigma }}\varphi \left({\frac {x-\mu }{\sigma }}\right)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29ad1537690c6dca78c0a0834983bcd08c085aaf" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.176ex; height:6.176ex;" alt="{\displaystyle f(x\mid \mu ,\sigma ^{2})={\frac {1}{\sigma }}\varphi \left({\frac {x-\mu }{\sigma }}\right)}"></span></dd></dl>
<p>The probability density must be scaled by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/\sigma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>&#x03C3;<!-- σ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 1/\sigma }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00d5187486468042e9692b18c216b60679aafef3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.655ex; height:2.843ex;" alt="1/\sigma "></span> so that the <a href="/wiki/Integral" title="Integral">integral</a> is still&#160;1.
</p><p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Z</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="Z"></span> is a <a href="/wiki/Standard_normal_deviate" title="Standard normal deviate">standard normal deviate</a>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=\sigma Z+\mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>=</mo>
<mi>&#x03C3;<!-- σ --></mi>
<mi>Z</mi>
<mo>+</mo>
<mi>&#x03BC;<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X=\sigma Z+\mu }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa7a3a0442e6e0db7264d42886c76cf7e16bc77a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.331ex; height:2.676ex;" alt="{\displaystyle X=\sigma Z+\mu }"></span> will have a normal distribution with expected value <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="\mu "></span> and standard deviation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03C3;<!-- σ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="\sigma "></span>. This is equivalent to saying that the standard normal distribution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Z</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="Z"></span> can be scaled/stretched by a factor of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03C3;<!-- σ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="\sigma "></span> and shifted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="\mu "></span> to yield a different normal distribution, called <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"></span>. Conversely, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"></span> is a normal deviate with parameters <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="\mu "></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="\sigma ^{2}"></span>, then this <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"></span> distribution can be re-scaled and shifted via the formula <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z=(X-\mu )/\sigma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Z</mi>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>&#x03C3;<!-- σ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Z=(X-\mu )/\sigma }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25037543a35e0e5689bd9bf285d59e083e5382aa" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.302ex; height:2.843ex;" alt="{\displaystyle Z=(X-\mu )/\sigma }"></span> to convert it to the standard normal distribution. This variate is also called the standardized form of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"></span>.
</p>
<h3><span class="mw-headline" id="Notation">Notation</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=4" title="Edit section: Notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
<p>The probability density of the standard Gaussian distribution (standard normal distribution, with zero mean and unit variance) is often denoted with the Greek letter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03D5;<!-- ϕ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \phi }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="\phi "></span> (<a href="/wiki/Phi_(letter)" class="mw-redirect" title="Phi (letter)">phi</a>).<sup id="cite_ref-8" class="reference"><a href="#cite_note-8">&#91;8&#93;</a></sup> The alternative form of the Greek letter phi, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03C6;<!-- φ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="\varphi "></span>, is also used quite often.
</p><p>The normal distribution is often referred to as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(\mu ,\sigma ^{2})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>N</mi>
<mo stretchy="false">(</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>,</mo>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle N(\mu ,\sigma ^{2})}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cbd76720b12f0428a8bf1460b7a67cf2f5f3817" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.693ex; height:3.176ex;" alt="N(\mu ,\sigma ^{2})"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}(\mu ,\sigma ^{2})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">N</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>,</mo>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}(\mu ,\sigma ^{2})}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/863304aaa42a945f2f07d79facc3d2eebc845ce7" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.062ex; width:8.966ex; height:3.176ex;" alt="{\mathcal {N}}(\mu ,\sigma ^{2})"></span>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9">&#91;9&#93;</a></sup> Thus when a random variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"></span> is normally distributed with mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="\mu "></span> and standard deviation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03C3;<!-- σ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="\sigma "></span>, one may write
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim {\mathcal {N}}(\mu ,\sigma ^{2}).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>&#x223C;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">N</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>,</mo>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X\sim {\mathcal {N}}(\mu ,\sigma ^{2}).}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0aeea1216143061c89f6a1944928a0aeee1b9cb1" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.63ex; height:3.176ex;" alt="{\displaystyle X\sim {\mathcal {N}}(\mu ,\sigma ^{2}).}"></span></dd></dl>
<h3><span class="mw-headline" id="Alternative_parameterizations">Alternative parameterizations</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=5" title="Edit section: Alternative parameterizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
<p>Some authors advocate using the <a href="/wiki/Precision_(statistics)" title="Precision (statistics)">precision</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03C4;<!-- τ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \tau }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="\tau "></span> as the parameter defining the width of the distribution, instead of the deviation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03C3;<!-- σ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="\sigma "></span> or the variance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="\sigma ^{2}"></span>. The precision is normally defined as the reciprocal of the variance, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/\sigma ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 1/\sigma ^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd9d6d70944c9de586516f90477d752079617c07" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.71ex; height:3.176ex;" alt="{\displaystyle 1/\sigma ^{2}}"></span>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10">&#91;10&#93;</a></sup> The formula for the distribution then becomes
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\sqrt {\frac {\tau }{2\pi }}}e^{-\tau (x-\mu )^{2}/2}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mfrac>
<mi>&#x03C4;<!-- τ --></mi>
<mrow>
<mn>2</mn>
<mi>&#x03C0;<!-- π --></mi>
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</mrow>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<mi>&#x03C4;<!-- τ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
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<mn>2</mn>
</mrow>
</msup>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(x)={\sqrt {\frac {\tau }{2\pi }}}e^{-\tau (x-\mu )^{2}/2}.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e18260c517de859f1451477bc0c91d2d46f092a1" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.226ex; height:6.343ex;" alt="{\displaystyle f(x)={\sqrt {\frac {\tau }{2\pi }}}e^{-\tau (x-\mu )^{2}/2}.}"></span></dd></dl>
<p>This choice is claimed to have advantages in numerical computations when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03C3;<!-- σ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="\sigma "></span> is very close to zero, and simplifies formulas in some contexts, such as in the <a href="/wiki/Bayesian_statistics" title="Bayesian statistics">Bayesian inference</a> of variables with <a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">multivariate normal distribution</a>.
</p><p>Alternatively, the reciprocal of the standard deviation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau '=1/\sigma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>&#x03C4;<!-- τ --></mi>
<mo>&#x2032;</mo>
</msup>
<mo>=</mo>
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>&#x03C3;<!-- σ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \tau '=1/\sigma }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b39414538a7759f87026fcdf699b1f5501d8803d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.696ex; height:3.009ex;" alt="{\displaystyle \tau &#039;=1/\sigma }"></span> might be defined as the <i>precision</i>, in which case the expression of the normal distribution becomes
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\frac {\tau '}{\sqrt {2\pi }}}e^{-(\tau ')^{2}(x-\mu )^{2}/2}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>&#x03C4;<!-- τ --></mi>
<mo>&#x2032;</mo>
</msup>
<msqrt>
<mn>2</mn>
<mi>&#x03C0;<!-- π --></mi>
</msqrt>
</mfrac>
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<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<mo stretchy="false">(</mo>
<msup>
<mi>&#x03C4;<!-- τ --></mi>
<mo>&#x2032;</mo>
</msup>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
</mrow>
</msup>
<mo>.</mo>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {\tau '}{\sqrt {2\pi }}}e^{-(\tau ')^{2}(x-\mu )^{2}/2}.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e16b2715bf35fcc08b305f756ca921d8557ddc4f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:26.52ex; height:6.509ex;" alt="{\displaystyle f(x)={\frac {\tau &#039;}{\sqrt {2\pi }}}e^{-(\tau &#039;)^{2}(x-\mu )^{2}/2}.}"></span></dd></dl>
<p>According to Stigler, this formulation is advantageous because of a much simpler and easier-to-remember formula, and simple approximate formulas for the <a href="/wiki/Quantile" title="Quantile">quantiles</a> of the distribution.
</p><p>Normal distributions form an <a href="/wiki/Exponential_family" title="Exponential family">exponential family</a> with <a href="/wiki/Natural_parameter" class="mw-redirect" title="Natural parameter">natural parameters</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \theta _{1}={\frac {\mu }{\sigma ^{2}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mstyle displaystyle="false" scriptlevel="0">
<msub>
<mi>&#x03B8;<!-- θ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>&#x03BC;<!-- μ --></mi>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mfrac>
</mrow>
</mstyle>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \textstyle \theta _{1}={\frac {\mu }{\sigma ^{2}}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d82f6e67d857e20baab7aa27a4240120f4e14230" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:7.852ex; height:3.843ex;" alt="{\displaystyle \textstyle \theta _{1}={\frac {\mu }{\sigma ^{2}}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \theta _{2}={\frac {-1}{2\sigma ^{2}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mstyle displaystyle="false" scriptlevel="0">
<msub>
<mi>&#x03B8;<!-- θ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
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<mrow>
<mn>2</mn>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \textstyle \theta _{2}={\frac {-1}{2\sigma ^{2}}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0108435f9dbca84d4c70068b2e466606423e6368" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:8.674ex; height:4.009ex;" alt="{\displaystyle \textstyle \theta _{2}={\frac {-1}{2\sigma ^{2}}}}"></span>, and natural statistics <i>x</i> and <i>x</i><sup>2</sup>. The dual expectation parameters for normal distribution are <span class="nowrap"><i>η</i><sub>1</sub> = <i>μ</i></span> and <span class="nowrap"><i>η</i><sub>2</sub> = <i>μ</i><sup>2</sup> + <i>σ</i><sup>2</sup></span>.
</p>
<h3><span class="mw-headline" id="Cumulative_distribution_function">Cumulative distribution function</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=6" title="Edit section: Cumulative distribution function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
<p>The <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> (CDF) of the standard normal distribution, usually denoted with the capital Greek letter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="\Phi "></span> (<a href="/wiki/Phi_(letter)" class="mw-redirect" title="Phi (letter)">phi</a>), is the integral
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (x)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x}e^{-t^{2}/2}\,dt}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<msqrt>
<mn>2</mn>
<mi>&#x03C0;<!-- π --></mi>
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<msubsup>
<mo>&#x222B;<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<mi mathvariant="normal">&#x221E;<!-- ∞ --></mi>
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<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msup>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
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</msup>
<mspace width="thinmathspace" />
<mi>d</mi>
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<annotation encoding="application/x-tex">{\displaystyle \Phi (x)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x}e^{-t^{2}/2}\,dt}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbbfde462f3432f932e7bc59a5f7351c0349d094" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:26.509ex; height:6.343ex;" alt="{\displaystyle \Phi (x)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x}e^{-t^{2}/2}\,dt}"></span></dd></dl>
<h3><span class="mw-headline" id="Error_Function">Error Function</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=7" title="Edit section: Error Function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
<p>The related <a href="/wiki/Error_function" title="Error function">error function</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {erf} (x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>erf</mi>
<mo>&#x2061;<!-- --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {erf} (x)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2a96a1b139214ea50c6d6f436fb555e6429134e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.795ex; height:2.843ex;" alt="\operatorname{erf}(x)"></span> gives the probability of a random variable, with normal distribution of mean 0 and variance 1/2 falling in the range <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [-x,x]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<mo>&#x2212;<!-- --></mo>
<mi>x</mi>
<mo>,</mo>
<mi>x</mi>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle [-x,x]}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e23c41ff0bd6f01a0e27054c2b85819fcd08b762" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.795ex; height:2.843ex;" alt="[-x,x]"></span>. That is:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {erf} (x)={\frac {1}{\sqrt {\pi }}}\int _{-x}^{x}e^{-t^{2}}\,dt={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{-t^{2}}\,dt}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>erf</mi>
<mo>&#x2061;<!-- --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<msqrt>
<mi>&#x03C0;<!-- π --></mi>
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<msubsup>
<mo>&#x222B;<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<mi>x</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
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</msubsup>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</msup>
<mspace width="thinmathspace" />
<mi>d</mi>
<mi>t</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>2</mn>
<msqrt>
<mi>&#x03C0;<!-- π --></mi>
</msqrt>
</mfrac>
</mrow>
<msubsup>
<mo>&#x222B;<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
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</msubsup>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</msup>
<mspace width="thinmathspace" />
<mi>d</mi>
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {erf} (x)={\frac {1}{\sqrt {\pi }}}\int _{-x}^{x}e^{-t^{2}}\,dt={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{-t^{2}}\,dt}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7851781b8f281948dc1cb2d8c36b01bada99122" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:42.053ex; height:6.343ex;" alt="{\displaystyle \operatorname {erf} (x)={\frac {1}{\sqrt {\pi }}}\int _{-x}^{x}e^{-t^{2}}\,dt={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{-t^{2}}\,dt}"></span></dd></dl>
<p>These integrals cannot be expressed in terms of elementary functions, and are often said to be <a href="/wiki/Special_function" class="mw-redirect" title="Special function">special functions</a>. However, many numerical approximations are known; see <a href="#Numerical_approximations_for_the_normal_cumulative_distribution_function_and_normal_quantile_function">below</a> for more.
</p><p>The two functions are closely related, namely
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (x)={\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)\right]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mrow>
<mo>[</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>erf</mi>
<mo>&#x2061;<!-- --></mo>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>x</mi>
<msqrt>
<mn>2</mn>
</msqrt>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Phi (x)={\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)\right]}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7831a9a5f630df7170fa805c186f4c53219ca36" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:26.771ex; height:6.509ex;" alt="{\displaystyle \Phi (x)={\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)\right]}"></span></dd></dl>
<p>For a generic normal distribution with density <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="f"></span>, mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="\mu "></span> and deviation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03C3;<!-- σ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="\sigma "></span>, the cumulative distribution function is
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)=\Phi \left({\frac {x-\mu }{\sigma }}\right)={\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)\right]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>x</mi>
<mo>&#x2212;<!-- --></mo>
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</mrow>
<mi>&#x03C3;<!-- σ --></mi>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
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<mo>[</mo>
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<mn>1</mn>
<mo>+</mo>
<mi>erf</mi>
<mo>&#x2061;<!-- --></mo>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>x</mi>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
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<mrow>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>2</mn>
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<mo>)</mo>
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<mo>]</mo>
</mrow>
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<annotation encoding="application/x-tex">{\displaystyle F(x)=\Phi \left({\frac {x-\mu }{\sigma }}\right)={\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)\right]}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75deccfbc473d782dacb783f1524abb09b8135c0" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:44.299ex; height:6.509ex;" alt="{\displaystyle F(x)=\Phi \left({\frac {x-\mu }{\sigma }}\right)={\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)\right]}"></span></dd></dl>
<p>The complement of the standard normal cumulative distribution function, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(x)=1-\Phi (x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Q</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>1</mn>
<mo>&#x2212;<!-- --></mo>
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Q(x)=1-\Phi (x)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8003a26bddf37fc2d651514d6f4662c0de39aff3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.896ex; height:2.843ex;" alt="Q(x)=1-\Phi (x)"></span>, is often called the <a href="/wiki/Q-function" title="Q-function">Q-function</a>, especially in engineering texts.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11">&#91;11&#93;</a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12">&#91;12&#93;</a></sup> It gives the probability that the value of a standard normal random variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"></span> will exceed <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="x"></span>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(X&gt;x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>P</mi>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>&gt;</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P(X&gt;x)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/767fd276524cfb3556093722a4f40a9209194ea5" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.963ex; height:2.843ex;" alt="{\displaystyle P(X&gt;x)}"></span>. Other definitions of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Q</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Q}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="Q"></span>-function, all of which are simple transformations of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="\Phi "></span>, are also used occasionally.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13">&#91;13&#93;</a></sup>
</p><p>The <a href="/wiki/Graph_of_a_function" title="Graph of a function">graph</a> of the standard normal cumulative distribution function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="\Phi "></span> has 2-fold <a href="/wiki/Rotational_symmetry" title="Rotational symmetry">rotational symmetry</a> around the point (0,1/2); that is, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (-x)=1-\Phi (x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
<mo stretchy="false">(</mo>
<mo>&#x2212;<!-- --></mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>1</mn>
<mo>&#x2212;<!-- --></mo>
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Phi (-x)=1-\Phi (x)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac1a5e4fc7858485f2a5448635fd0a85b7fd53b0" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.544ex; height:2.843ex;" alt="\Phi (-x)=1-\Phi (x)"></span>. Its <a href="/wiki/Antiderivative" title="Antiderivative">antiderivative</a> (indefinite integral) can be expressed as follows:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \Phi (x)\,dx=x\Phi (x)+\varphi (x)+C.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>&#x222B;<!-- ∫ --></mo>
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace" />
<mi>d</mi>
<mi>x</mi>
<mo>=</mo>
<mi>x</mi>
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>&#x03C6;<!-- φ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>C</mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \int \Phi (x)\,dx=x\Phi (x)+\varphi (x)+C.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ec7f747ce873d091260c617c82359d7c407fee6" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.329ex; height:5.676ex;" alt="{\displaystyle \int \Phi (x)\,dx=x\Phi (x)+\varphi (x)+C.}"></span></dd></dl>
<p>The cumulative distribution function of the standard normal distribution can be expanded by <a href="/wiki/Integration_by_parts" title="Integration by parts">Integration by parts</a> into a series:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (x)={\frac {1}{2}}+{\frac {1}{\sqrt {2\pi }}}\cdot e^{-x^{2}/2}\left[x+{\frac {x^{3}}{3}}+{\frac {x^{5}}{3\cdot 5}}+\cdots +{\frac {x^{2n+1}}{(2n+1)!!}}+\cdots \right]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
<mo stretchy="false">(</mo>
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<mo>=</mo>
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<mi>e</mi>
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<mi>x</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
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<mi>x</mi>
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</msup>
<mn>3</mn>
</mfrac>
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<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>5</mn>
</mrow>
</msup>
<mrow>
<mn>3</mn>
<mo>&#x22C5;<!-- ⋅ --></mo>
<mn>5</mn>
</mrow>
</mfrac>
</mrow>
<mo>+</mo>
<mo>&#x22EF;<!-- ⋯ --></mo>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
<mrow>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>!</mo>
<mo>!</mo>
</mrow>
</mfrac>
</mrow>
<mo>+</mo>
<mo>&#x22EF;<!-- ⋯ --></mo>
</mrow>
<mo>]</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Phi (x)={\frac {1}{2}}+{\frac {1}{\sqrt {2\pi }}}\cdot e^{-x^{2}/2}\left[x+{\frac {x^{3}}{3}}+{\frac {x^{5}}{3\cdot 5}}+\cdots +{\frac {x^{2n+1}}{(2n+1)!!}}+\cdots \right]}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54d12af9a3b12a7f859e4e7be105d172b53bcfb8" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:68.09ex; height:6.676ex;" alt="{\displaystyle \Phi (x)={\frac {1}{2}}+{\frac {1}{\sqrt {2\pi }}}\cdot e^{-x^{2}/2}\left[x+{\frac {x^{3}}{3}}+{\frac {x^{5}}{3\cdot 5}}+\cdots +{\frac {x^{2n+1}}{(2n+1)!!}}+\cdots \right]}"></span></dd></dl>
<p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle !!}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>!</mo>
<mo>!</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle !!}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a6e2480ece878ba9a96d09f1fe710c7117f82f8" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.294ex; height:2.009ex;" alt="!!"></span> denotes the <a href="/wiki/Double_factorial" title="Double factorial">double factorial</a>.
</p><p>An <a href="/wiki/Asymptotic_expansion" title="Asymptotic expansion">asymptotic expansion</a> of the cumulative distribution function for large <i>x</i> can also be derived using integration by parts. For more, see <a href="/wiki/Error_function#Asymptotic_expansion" title="Error function">Error function#Asymptotic expansion</a>.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14">&#91;14&#93;</a></sup>
</p><p>A quick approximation to the standard normal distribution's cumulative distribution function can be found by using a Taylor series approximation:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (x)\approx {\frac {1}{2}}+{\frac {1}{\sqrt {2\pi }}}\sum _{k=0}^{n}{\frac {(-1)^{k}x^{(2k+1)}}{2^{k}k!(2k+1)}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>&#x2248;<!-- ≈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
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<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<msqrt>
<mn>2</mn>
<mi>&#x03C0;<!-- π --></mi>
</msqrt>
</mfrac>
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<munderover>
<mo>&#x2211;<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>=</mo>
<mn>0</mn>
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<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
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<annotation encoding="application/x-tex">{\displaystyle \Phi (x)\approx {\frac {1}{2}}+{\frac {1}{\sqrt {2\pi }}}\sum _{k=0}^{n}{\frac {(-1)^{k}x^{(2k+1)}}{2^{k}k!(2k+1)}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff6b356abee5bef96908b11cc4533be515ddc145" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:35.475ex; height:7.343ex;" alt="{\displaystyle \Phi (x)\approx {\frac {1}{2}}+{\frac {1}{\sqrt {2\pi }}}\sum _{k=0}^{n}{\frac {(-1)^{k}x^{(2k+1)}}{2^{k}k!(2k+1)}}}"></span></dd></dl>
<h4><span class="mw-headline" id="Recursive_computation_with_Taylor_series_expansion">Recursive computation with Taylor series expansion</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=8" title="Edit section: Recursive computation with Taylor series expansion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h4>
<p>The recursive nature of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{ax^{2}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>a</mi>
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<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle e^{ax^{2}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91be4eeabff58ba2dfc5ac9358433ee6111aa46e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.957ex; height:3.009ex;" alt="e^{ax^{2}}"></span>family of derivatives may be used to easily construct a rapidly converging <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a> expansion using recursive entries about any point of known value of the distribution,<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (x_{0})}">
<semantics>
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<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \Phi (x_{0})}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c40d0a76821f94e6ba23d25e9a78553878e3611a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.871ex; height:2.843ex;" alt="{\displaystyle \Phi (x_{0})}"></span>:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (x)=\sum _{n=0}^{\infty }{\frac {\Phi ^{(n)}(x_{0})}{n!}}(x-x_{0})^{n}}">
<semantics>
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<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle \Phi (x)=\sum _{n=0}^{\infty }{\frac {\Phi ^{(n)}(x_{0})}{n!}}(x-x_{0})^{n}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f36fea556eb12345f32c96c70aeb28b6b3cf1cc3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.445ex; height:7.176ex;" alt="{\displaystyle \Phi (x)=\sum _{n=0}^{\infty }{\frac {\Phi ^{(n)}(x_{0})}{n!}}(x-x_{0})^{n}}"></span></dd></dl>
<p>where:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi ^{(0)}(x_{0})={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x_{0}}e^{-t^{2}/2}\,dt}">
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<annotation encoding="application/x-tex">{\displaystyle \Phi ^{(0)}(x_{0})={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x_{0}}e^{-t^{2}/2}\,dt}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0a2e15a4a7401eb5ac46e8ec1b7ddf3384df133" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:29.918ex; height:6.343ex;" alt="{\displaystyle \Phi ^{(0)}(x_{0})={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x_{0}}e^{-t^{2}/2}\,dt}"></span></dd></dl>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi ^{(1)}(x_{0})={\frac {1}{\sqrt {2\pi }}}e^{-x_{0}^{2}/2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
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<mo stretchy="false">(</mo>
<mn>1</mn>
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<mi>e</mi>
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<annotation encoding="application/x-tex">{\displaystyle \Phi ^{(1)}(x_{0})={\frac {1}{\sqrt {2\pi }}}e^{-x_{0}^{2}/2}}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebbf5f699085abcfea6ee6e2feec52096188ba3d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:22.58ex; height:6.176ex;" alt="{\displaystyle \Phi ^{(1)}(x_{0})={\frac {1}{\sqrt {2\pi }}}e^{-x_{0}^{2}/2}}"></span></dd></dl>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi ^{(n)}(x_{0})=-(x_{0}\Phi ^{(n-1)}(x_{0})+(n-2)\Phi ^{(n-2)}(x_{0}))}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
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<mo stretchy="false">(</mo>
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<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mi>n</mi>
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<mo stretchy="false">)</mo>
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<annotation encoding="application/x-tex">{\displaystyle \Phi ^{(n)}(x_{0})=-(x_{0}\Phi ^{(n-1)}(x_{0})+(n-2)\Phi ^{(n-2)}(x_{0}))}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfb5e486ef1429af8a44bbfc43a68c2317d0f5da" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.456ex; height:3.343ex;" alt="{\displaystyle \Phi ^{(n)}(x_{0})=-(x_{0}\Phi ^{(n-1)}(x_{0})+(n-2)\Phi ^{(n-2)}(x_{0}))}"></span>, for all <i>n</i> ≥ 2.</dd></dl>
<h4><span id="Using_the_Taylor_series_and_Newton.27s_method_for_the_inverse_function"></span><span class="mw-headline" id="Using_the_Taylor_series_and_Newton's_method_for_the_inverse_function">Using the Taylor series and Newton's method for the inverse function</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=9" title="Edit section: Using the Taylor series and Newton&#039;s method for the inverse function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h4>
<p>An application for the above <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a> expansion is to use <a href="/wiki/Newton%27s_method" title="Newton&#39;s method">Newton's method</a> to reverse the computation. That is, if we have a value for the <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Phi (x)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79e4f01c93494fbb5dcd75761f4468121b00b294" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.817ex; height:2.843ex;" alt="\Phi (x)"></span>, but do not know the x needed to obtain the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Phi (x)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79e4f01c93494fbb5dcd75761f4468121b00b294" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.817ex; height:2.843ex;" alt="\Phi (x)"></span>, we can use Newton's method to find x, and use the Taylor series expansion above to minimize the number of computations. Newton's method is ideal to solve this problem because the first derivative of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Phi (x)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79e4f01c93494fbb5dcd75761f4468121b00b294" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.817ex; height:2.843ex;" alt="\Phi (x)"></span>, which is an integral of the normal standard distribution, is the normal standard distribution, and is readily available to use in the Newton's method solution.
</p><p>To solve, select a known approximate solution, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="x_{0}"></span>, to the desired <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Phi (x)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79e4f01c93494fbb5dcd75761f4468121b00b294" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.817ex; height:2.843ex;" alt="\Phi (x)"></span>. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="x_{0}"></span> may be a value from a distribution table, or an intelligent estimate followed by a computation of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (x_{0})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Phi (x_{0})}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c40d0a76821f94e6ba23d25e9a78553878e3611a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.871ex; height:2.843ex;" alt="{\displaystyle \Phi (x_{0})}"></span> using any desired means to compute. Use this value of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="x_{0}"></span> and the Taylor series expansion above to minimize computations.
</p><p>Repeat the following process until the difference between the computed <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (x_{n})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Phi (x_{n})}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b0625d133f5473a4001f4337252866d822e88c7" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.036ex; height:2.843ex;" alt="{\displaystyle \Phi (x_{n})}"></span> and the desired <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="\Phi "></span>, which we will call <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi ({\text{desired}})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>desired</mtext>
</mrow>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Phi ({\text{desired}})}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4eb0149802066ceef655e928be452fd191d02ac6" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.612ex; height:2.843ex;" alt="{\displaystyle \Phi ({\text{desired}})}"></span>, is below a chosen acceptably small error, such as 10<sup>5</sup>, 10<sup>15</sup>, etc.:
</p><p><div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n+1}=x_{n}-{\frac {\Phi (x_{n},x_{0},\Phi (x_{0}))-\Phi ({\text{desired}})}{\Phi '(x_{n})}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>,</mo>
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>&#x2212;<!-- --></mo>
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>desired</mtext>
</mrow>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<msup>
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
<mo>&#x2032;</mo>
</msup>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{n+1}=x_{n}-{\frac {\Phi (x_{n},x_{0},\Phi (x_{0}))-\Phi ({\text{desired}})}{\Phi '(x_{n})}}}</annotation>
</semantics>
</math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f50ddc0b1d5d705e21fa7be9b550091498fb0b" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:43.783ex; height:6.509ex;" alt="{\displaystyle x_{n+1}=x_{n}-{\frac {\Phi (x_{n},x_{0},\Phi (x_{0}))-\Phi ({\text{desired}})}{\Phi &#039;(x_{n})}}}"></div>
</p><p>where
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (x,x_{0},\Phi (x_{0}))}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>,</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>,</mo>
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Phi (x,x_{0},\Phi (x_{0}))}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9676d57afa6edf8c0b3e67df4b69faf4e5fedcb7" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.14ex; height:2.843ex;" alt="{\displaystyle \Phi (x,x_{0},\Phi (x_{0}))}"></span> is the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Phi (x)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79e4f01c93494fbb5dcd75761f4468121b00b294" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.817ex; height:2.843ex;" alt="\Phi (x)"></span> from a Taylor series solution using <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="x_{0}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (x_{0})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Phi (x_{0})}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c40d0a76821f94e6ba23d25e9a78553878e3611a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.871ex; height:2.843ex;" alt="{\displaystyle \Phi (x_{0})}"></span></dd></dl>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi '(x)={\frac {1}{\sqrt {2\pi }}}e^{-x^{2}/2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
<mo>&#x2032;</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<msqrt>
<mn>2</mn>
<mi>&#x03C0;<!-- π --></mi>
</msqrt>
</mfrac>
</mrow>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Phi '(x)={\frac {1}{\sqrt {2\pi }}}e^{-x^{2}/2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7de0b355ff110547f65d31cbe403ff646272819f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:19.877ex; height:6.176ex;" alt="{\displaystyle \Phi &#039;(x)={\frac {1}{\sqrt {2\pi }}}e^{-x^{2}/2}}"></span></dd></dl>
<p>When the repeated computations converge to an error below the chosen acceptably small value, <i>x</i> will be the value needed to obtain a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Phi (x)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79e4f01c93494fbb5dcd75761f4468121b00b294" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.817ex; height:2.843ex;" alt="\Phi (x)"></span> of the desired value, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi ({\text{desired}})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mtext>desired</mtext>
</mrow>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Phi ({\text{desired}})}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4eb0149802066ceef655e928be452fd191d02ac6" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.612ex; height:2.843ex;" alt="{\displaystyle \Phi ({\text{desired}})}"></span>. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="x_{0}"></span> is a good beginning estimate, convergence should be rapid with only a small number of iterations needed.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">&#91;<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="Section does not cite any sources (April 2023)">citation needed</span></a></i>&#93;</sup>
</p>
<h4><span class="mw-headline" id="Standard_deviation_and_coverage">Standard deviation and coverage</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=10" title="Edit section: Standard deviation and coverage"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h4>
<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1033289096"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Interval_estimation" title="Interval estimation">Interval estimation</a> and <a href="/wiki/Coverage_probability" title="Coverage probability">Coverage probability</a></div>
<figure typeof="mw:File/Thumb"><a href="/wiki/File:Standard_deviation_diagram.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Standard_deviation_diagram.svg/350px-Standard_deviation_diagram.svg.png" decoding="async" width="350" height="175" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Standard_deviation_diagram.svg/525px-Standard_deviation_diagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Standard_deviation_diagram.svg/700px-Standard_deviation_diagram.svg.png 2x" data-file-width="400" data-file-height="200" /></a><figcaption>For the normal distribution, the values less than one standard deviation away from the mean account for 68.27% of the set; while two standard deviations from the mean account for 95.45%; and three standard deviations account for 99.73%.</figcaption></figure>
<p>About 68% of values drawn from a normal distribution are within one standard deviation <i>σ</i> away from the mean; about 95% of the values lie within two standard deviations; and about 99.7% are within three standard deviations.<sup id="cite_ref-mathsisfun_6-1" class="reference"><a href="#cite_note-mathsisfun-6">&#91;6&#93;</a></sup> This fact is known as the <a href="/wiki/68%E2%80%9395%E2%80%9399.7_rule" title="689599.7 rule">68-95-99.7 (empirical) rule</a>, or the <i>3-sigma rule</i>.
</p><p>More precisely, the probability that a normal deviate lies in the range between <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu -n\sigma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
<mo>&#x2212;<!-- --></mo>
<mi>n</mi>
<mi>&#x03C3;<!-- σ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu -n\sigma }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbdf9b56523a7d1bc130894cdbca89c096c61a3b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.966ex; height:2.509ex;" alt="{\displaystyle \mu -n\sigma }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu +n\sigma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
<mo>+</mo>
<mi>n</mi>
<mi>&#x03C3;<!-- σ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu +n\sigma }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4fc47206a6838c8b99e0f16cce4fafd5f8a37c" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.966ex; height:2.509ex;" alt="{\displaystyle \mu +n\sigma }"></span> is given by
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(\mu +n\sigma )-F(\mu -n\sigma )=\Phi (n)-\Phi (-n)=\operatorname {erf} \left({\frac {n}{\sqrt {2}}}\right).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>+</mo>
<mi>n</mi>
<mi>&#x03C3;<!-- σ --></mi>
<mo stretchy="false">)</mo>
<mo>&#x2212;<!-- --></mo>
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>&#x2212;<!-- --></mo>
<mi>n</mi>
<mi>&#x03C3;<!-- σ --></mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>&#x2212;<!-- --></mo>
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
<mo stretchy="false">(</mo>
<mo>&#x2212;<!-- --></mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>erf</mi>
<mo>&#x2061;<!-- --></mo>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>n</mi>
<msqrt>
<mn>2</mn>
</msqrt>
</mfrac>
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<mo>)</mo>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F(\mu +n\sigma )-F(\mu -n\sigma )=\Phi (n)-\Phi (-n)=\operatorname {erf} \left({\frac {n}{\sqrt {2}}}\right).}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/effeceb477bf37b05d0035347946350b1f0155ce" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:55.141ex; height:6.509ex;" alt="{\displaystyle F(\mu +n\sigma )-F(\mu -n\sigma )=\Phi (n)-\Phi (-n)=\operatorname {erf} \left({\frac {n}{\sqrt {2}}}\right).}"></span></dd></dl>
<p>To 12 significant digits, the values for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=1,2,\ldots ,6}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mn>2</mn>
<mo>,</mo>
<mo>&#x2026;<!-- … --></mo>
<mo>,</mo>
<mn>6</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n=1,2,\ldots ,6}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e82d12a97f526d2a6ce41dff3f71e5bf4f38bd3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.193ex; height:2.509ex;" alt="{\displaystyle n=1,2,\ldots ,6}"></span> are:<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">&#91;<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (August 2022)">citation needed</span></a></i>&#93;</sup>
</p>
<table class="wikitable" style="text-align:center;margin-left:24pt">
<tbody><tr>
<th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="n"></span></th>
<th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=F(\mu +n\sigma )-F(\mu -n\sigma )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
<mo>=</mo>
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>+</mo>
<mi>n</mi>
<mi>&#x03C3;<!-- σ --></mi>
<mo stretchy="false">)</mo>
<mo>&#x2212;<!-- --></mo>
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>&#x2212;<!-- --></mo>
<mi>n</mi>
<mi>&#x03C3;<!-- σ --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p=F(\mu +n\sigma )-F(\mu -n\sigma )}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15d505d55c4fd7f494215ab2b35caa3ffd07eae6" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:28.231ex; height:2.843ex;" alt="{\displaystyle p=F(\mu +n\sigma )-F(\mu -n\sigma )}"></span></th>
<th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{i.e. }}1-p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtext>i.e.&#xA0;</mtext>
</mrow>
<mn>1</mn>
<mo>&#x2212;<!-- --></mo>
<mi>p</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\text{i.e. }}1-p}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3047a02e84df20b65d8a4a7022c999097413d03e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.726ex; height:2.509ex;" alt="{\displaystyle {\text{i.e. }}1-p}"></span></th>
<th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{or }}1{\text{ in }}p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtext>or&#xA0;</mtext>
</mrow>
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mtext>&#xA0;in&#xA0;</mtext>
</mrow>
<mi>p</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\text{or }}1{\text{ in }}p}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50166c3fb485e51c33ff391e9cdc31dd59d185ec" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.087ex; height:2.509ex;" alt="{\displaystyle {\text{or }}1{\text{ in }}p}"></span></th>
<th><a href="/wiki/OEIS" class="mw-redirect" title="OEIS">OEIS</a>
</th></tr>
<tr>
<td>1</td>
<td><span class="nowrap"><span data-sort-value="6999682689492137000♠"></span>0.682<span style="margin-left:.25em;">689</span><span style="margin-left:.25em;">492</span><span style="margin-left:.25em;">137</span></span></td>
<td><span class="nowrap"><span data-sort-value="6999317310507863000♠"></span>0.317<span style="margin-left:.25em;">310</span><span style="margin-left:.25em;">507</span><span style="margin-left:.25em;">863</span></span></td>
<td>
<table cellpadding="0" cellspacing="0" style="width: 16em;">
<tbody><tr>
<td style="text-align: right; width: 7em;"><span class="nowrap"><span data-sort-value="7000300000000000000♠"></span>3</span></td>
<td style="text-align: left; width: 9em;"><span style="white-space:nowrap">.151<span style="margin-left:.25em">487</span><span style="margin-left:.25em">187</span><span style="margin-left:.25em">53</span></span>
</td></tr></tbody></table>
</td>
<td><span class="nowrap external"><a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>:&#160;<a href="//oeis.org/A178647" class="extiw" title="oeis:A178647">A178647</a></span>
</td></tr>
<tr>
<td>2</td>
<td><span class="nowrap"><span data-sort-value="6999954499736104000♠"></span>0.954<span style="margin-left:.25em;">499</span><span style="margin-left:.25em;">736</span><span style="margin-left:.25em;">104</span></span></td>
<td><span class="nowrap"><span data-sort-value="6998455002638960000♠"></span>0.045<span style="margin-left:.25em;">500</span><span style="margin-left:.25em;">263</span><span style="margin-left:.25em;">896</span></span></td>
<td>
<table cellpadding="0" cellspacing="0" style="width: 16em;">
<tbody><tr>
<td style="text-align: right; width: 7em;"><span class="nowrap"><span data-sort-value="7001210000000000000♠"></span>21</span></td>
<td style="text-align: left; width: 9em;"><span style="white-space:nowrap">.977<span style="margin-left:.25em">894</span><span style="margin-left:.25em">5080</span></span>
</td></tr></tbody></table>
</td>
<td><span class="nowrap external"><a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>:&#160;<a href="//oeis.org/A110894" class="extiw" title="oeis:A110894">A110894</a></span>
</td></tr>
<tr>
<td>3</td>
<td><span class="nowrap"><span data-sort-value="6999997300203937000♠"></span>0.997<span style="margin-left:.25em;">300</span><span style="margin-left:.25em;">203</span><span style="margin-left:.25em;">937</span></span></td>
<td><span class="nowrap"><span data-sort-value="6997269979606300000♠"></span>0.002<span style="margin-left:.25em;">699</span><span style="margin-left:.25em;">796</span><span style="margin-left:.25em;">063</span></span></td>
<td>
<table cellpadding="0" cellspacing="0" style="width: 16em;">
<tbody><tr>
<td style="text-align: right; width: 7em;"><span class="nowrap"><span data-sort-value="7002370000000000000♠"></span>370</span></td>
<td style="text-align: left; width: 9em;"><span style="white-space:nowrap">.398<span style="margin-left:.25em">347</span><span style="margin-left:.25em">345</span></span>
</td></tr></tbody></table>
</td>
<td><span class="nowrap external"><a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>:&#160;<a href="//oeis.org/A270712" class="extiw" title="oeis:A270712">A270712</a></span>
</td></tr>
<tr>
<td>4</td>
<td><span class="nowrap"><span data-sort-value="6999999936657516000♠"></span>0.999<span style="margin-left:.25em;">936</span><span style="margin-left:.25em;">657</span><span style="margin-left:.25em;">516</span></span></td>
<td><span class="nowrap"><span data-sort-value="6995633424840000000♠"></span>0.000<span style="margin-left:.25em;">063</span><span style="margin-left:.25em;">342</span><span style="margin-left:.25em;">484</span></span></td>
<td>
<table cellpadding="0" cellspacing="0" style="width: 16em;">
<tbody><tr>
<td style="text-align: right; width: 7em;"><span class="nowrap"><span data-sort-value="7004157870000000000♠"></span>15<span style="margin-left:.25em;">787</span></span></td>
<td style="text-align: left; width: 9em;"><span style="white-space:nowrap">.192<span style="margin-left:.25em">7673</span></span>
</td></tr></tbody></table>
</td></tr>
<tr>
<td>5</td>
<td><span class="nowrap"><span data-sort-value="6999999999426697000♠"></span>0.999<span style="margin-left:.25em;">999</span><span style="margin-left:.25em;">426</span><span style="margin-left:.25em;">697</span></span></td>
<td><span class="nowrap"><span data-sort-value="6993573303000000000♠"></span>0.000<span style="margin-left:.25em;">000</span><span style="margin-left:.25em;">573</span><span style="margin-left:.25em;">303</span></span></td>
<td>
<table cellpadding="0" cellspacing="0" style="width: 16em;">
<tbody><tr>
<td style="text-align: right; width: 7em;"><span class="nowrap"><span data-sort-value="7006174427700000000♠"></span>1<span style="margin-left:.25em;">744</span><span style="margin-left:.25em;">277</span></span></td>
<td style="text-align: left; width: 9em;"><span style="white-space:nowrap">.893<span style="margin-left:.25em">62</span></span>
</td></tr></tbody></table>
</td></tr>
<tr>
<td>6</td>
<td><span class="nowrap"><span data-sort-value="6999999999998027000♠"></span>0.999<span style="margin-left:.25em;">999</span><span style="margin-left:.25em;">998</span><span style="margin-left:.25em;">027</span></span></td>
<td><span class="nowrap"><span data-sort-value="6991197300000000000♠"></span>0.000<span style="margin-left:.25em;">000</span><span style="margin-left:.25em;">001</span><span style="margin-left:.25em;">973</span></span></td>
<td>
<table cellpadding="0" cellspacing="0" style="width: 16em;">
<tbody><tr>
<td style="text-align: right; width: 7em;"><span class="nowrap"><span data-sort-value="7008506797345000000♠"></span>506<span style="margin-left:.25em;">797</span><span style="margin-left:.25em;">345</span></span></td>
<td style="text-align: left; width: 9em;"><span style="white-space:nowrap">.897</span>
</td></tr></tbody></table>
</td></tr></tbody></table>
<p>For large <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="n"></span>, one can use the approximation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-p\approx {\frac {e^{-n^{2}/2}}{n{\sqrt {\pi /2}}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>1</mn>
<mo>&#x2212;<!-- --></mo>
<mi>p</mi>
<mo>&#x2248;<!-- ≈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<msup>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
</mrow>
</msup>
<mrow>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>&#x03C0;<!-- π --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
</msqrt>
</mrow>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 1-p\approx {\frac {e^{-n^{2}/2}}{n{\sqrt {\pi /2}}}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/296630e925b7399d170e283ba414879ee3e72bd8" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:16.482ex; height:7.343ex;" alt="{\displaystyle 1-p\approx {\frac {e^{-n^{2}/2}}{n{\sqrt {\pi /2}}}}}"></span>.
</p>
<h4><span class="mw-headline" id="Quantile_function">Quantile function</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=11" title="Edit section: Quantile function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h4>
<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1033289096"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Quantile_function#Normal_distribution" title="Quantile function">Quantile function §&#160;Normal distribution</a></div>
<p>The <a href="/wiki/Quantile_function" title="Quantile function">quantile function</a> of a distribution is the inverse of the cumulative distribution function. The quantile function of the standard normal distribution is called the <a href="/wiki/Probit_function" class="mw-redirect" title="Probit function">probit function</a>, and can be expressed in terms of the inverse <a href="/wiki/Error_function" title="Error function">error function</a>:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi ^{-1}(p)={\sqrt {2}}\operatorname {erf} ^{-1}(2p-1),\quad p\in (0,1).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>p</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
<msup>
<mi>erf</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo>&#x2061;<!-- --></mo>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mi>p</mi>
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mspace width="1em" />
<mi>p</mi>
<mo>&#x2208;<!-- ∈ --></mo>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Phi ^{-1}(p)={\sqrt {2}}\operatorname {erf} ^{-1}(2p-1),\quad p\in (0,1).}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de61998182ddb364f8b77d67c1aa645685fb3c3b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.888ex; height:3.176ex;" alt="{\displaystyle \Phi ^{-1}(p)={\sqrt {2}}\operatorname {erf} ^{-1}(2p-1),\quad p\in (0,1).}"></span></dd></dl>
<p>For a normal random variable with mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="\mu "></span> and variance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="\sigma ^{2}"></span>, the quantile function is
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F^{-1}(p)=\mu +\sigma \Phi ^{-1}(p)=\mu +\sigma {\sqrt {2}}\operatorname {erf} ^{-1}(2p-1),\quad p\in (0,1).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>p</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>+</mo>
<mi>&#x03C3;<!-- σ --></mi>
<msup>
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>p</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>+</mo>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
<msup>
<mi>erf</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo>&#x2061;<!-- --></mo>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mi>p</mi>
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mspace width="1em" />
<mi>p</mi>
<mo>&#x2208;<!-- ∈ --></mo>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F^{-1}(p)=\mu +\sigma \Phi ^{-1}(p)=\mu +\sigma {\sqrt {2}}\operatorname {erf} ^{-1}(2p-1),\quad p\in (0,1).}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57ed565648bb5901c0da2dd3ad10b8d447d4c73c" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:61.256ex; height:3.176ex;" alt="{\displaystyle F^{-1}(p)=\mu +\sigma \Phi ^{-1}(p)=\mu +\sigma {\sqrt {2}}\operatorname {erf} ^{-1}(2p-1),\quad p\in (0,1).}"></span></dd></dl>
<p>The <a href="/wiki/Quantile" title="Quantile">quantile</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi ^{-1}(p)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>p</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Phi ^{-1}(p)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b7f80c7666db21c1f67f44d750e2f39e58efbff" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.99ex; height:3.176ex;" alt="\Phi ^{{-1}}(p)"></span> of the standard normal distribution is commonly denoted as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{p}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z_{p}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52498d5e243c71b94e48fa16217a3f4a17be6687" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.14ex; height:2.343ex;" alt="{\displaystyle z_{p}}"></span>. These values are used in <a href="/wiki/Hypothesis_testing" class="mw-redirect" title="Hypothesis testing">hypothesis testing</a>, construction of <a href="/wiki/Confidence_interval" title="Confidence interval">confidence intervals</a> and <a href="/wiki/Q%E2%80%93Q_plot" title="QQ plot">QQ plots</a>. A normal random variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"></span> will exceed <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu +z_{p}\sigma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
<mo>+</mo>
<msub>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
</mrow>
</msub>
<mi>&#x03C3;<!-- σ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu +z_{p}\sigma }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d06b2ba266e725647bd8d7c64698761209f8da6d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.712ex; height:2.676ex;" alt="{\displaystyle \mu +z_{p}\sigma }"></span> with probability <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>1</mn>
<mo>&#x2212;<!-- --></mo>
<mi>p</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 1-p}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9633a8692121eedfa99cace406205e5d1511ef8d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.172ex; height:2.509ex;" alt="1-p"></span>, and will lie outside the interval <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \pm z_{p}\sigma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
<mo>&#x00B1;<!-- ± --></mo>
<msub>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
</mrow>
</msub>
<mi>&#x03C3;<!-- σ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu \pm z_{p}\sigma }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c273c8b7c1c6023c678307e6daae057cb315ece1" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.712ex; height:2.843ex;" alt="{\displaystyle \mu \pm z_{p}\sigma }"></span> with probability <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2(1-p)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>2</mn>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>&#x2212;<!-- --></mo>
<mi>p</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 2(1-p)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7403414204a8e5a6b889202992b9824f826cc72c" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.144ex; height:2.843ex;" alt="{\displaystyle 2(1-p)}"></span>. In particular, the quantile <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{0.975}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0.975</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z_{0.975}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc50a99c011834df5f9212e790ef6cd38818fb57" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.059ex; height:2.009ex;" alt="{\displaystyle z_{0.975}}"></span> is <a href="/wiki/1.96" class="mw-redirect" title="1.96">1.96</a>; therefore a normal random variable will lie outside the interval <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \pm 1.96\sigma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
<mo>&#x00B1;<!-- ± --></mo>
<mn>1.96</mn>
<mi>&#x03C3;<!-- σ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu \pm 1.96\sigma }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30c5ecf3dd3dbee9bbf36a86418dab6be8254bfc" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.706ex; height:2.676ex;" alt="\mu \pm 1.96\sigma "></span> in only 5% of cases.
</p><p>The following table gives the quantile <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{p}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z_{p}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52498d5e243c71b94e48fa16217a3f4a17be6687" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.14ex; height:2.343ex;" alt="{\displaystyle z_{p}}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"></span> will lie in the range <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \pm z_{p}\sigma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
<mo>&#x00B1;<!-- ± --></mo>
<msub>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
</mrow>
</msub>
<mi>&#x03C3;<!-- σ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu \pm z_{p}\sigma }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c273c8b7c1c6023c678307e6daae057cb315ece1" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.712ex; height:2.843ex;" alt="{\displaystyle \mu \pm z_{p}\sigma }"></span> with a specified probability <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="p"></span>. These values are useful to determine <a href="/wiki/Tolerance_interval" title="Tolerance interval">tolerance interval</a> for <a href="/wiki/Sample_mean_and_sample_covariance#Sample_mean" class="mw-redirect" title="Sample mean and sample covariance">sample averages</a> and other statistical <a href="/wiki/Estimator" title="Estimator">estimators</a> with normal (or <a href="/wiki/Asymptotic" class="mw-redirect" title="Asymptotic">asymptotically</a> normal) distributions.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">&#91;<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (August 2022)">citation needed</span></a></i>&#93;</sup> The following table shows <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}\operatorname {erf} ^{-1}(p)=\Phi ^{-1}\left({\frac {p+1}{2}}\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
<msup>
<mi>erf</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo>&#x2061;<!-- --></mo>
<mo stretchy="false">(</mo>
<mi>p</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
</mrow>
</msup>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>p</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}\operatorname {erf} ^{-1}(p)=\Phi ^{-1}\left({\frac {p+1}{2}}\right)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ff85c86fd6c5fee87c07357614971d2295549b0" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.379ex; height:6.176ex;" alt="{\displaystyle {\sqrt {2}}\operatorname {erf} ^{-1}(p)=\Phi ^{-1}\left({\frac {p+1}{2}}\right)}"></span>, not <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi ^{-1}(p)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>p</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Phi ^{-1}(p)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b7f80c7666db21c1f67f44d750e2f39e58efbff" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.99ex; height:3.176ex;" alt="\Phi ^{{-1}}(p)"></span> as defined above.
</p>
<table class="wikitable" style="text-align:left;margin-left:24pt;border:none;background:none;">
<tbody><tr>
<th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="p"></span></th>
<th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{p}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z_{p}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52498d5e243c71b94e48fa16217a3f4a17be6687" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.14ex; height:2.343ex;" alt="{\displaystyle z_{p}}"></span>
</th>
<td rowspan="8" style="border:none;background:none;">&#160;
</td>
<th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="p"></span></th>
<th><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z_{p}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z_{p}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52498d5e243c71b94e48fa16217a3f4a17be6687" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.14ex; height:2.343ex;" alt="{\displaystyle z_{p}}"></span>
</th></tr>
<tr>
<td>0.80</td>
<td><span class="nowrap"><span data-sort-value="7000128155156554500♠"></span>1.281<span style="margin-left:.25em;">551</span><span style="margin-left:.25em;">565</span><span style="margin-left:.25em;">545</span></span></td>
<td>0.999</td>
<td><span class="nowrap"><span data-sort-value="7000329052673149200♠"></span>3.290<span style="margin-left:.25em;">526</span><span style="margin-left:.25em;">731</span><span style="margin-left:.25em;">492</span></span>
</td></tr>
<tr>
<td>0.90</td>
<td><span class="nowrap"><span data-sort-value="7000164485362695100♠"></span>1.644<span style="margin-left:.25em;">853</span><span style="margin-left:.25em;">626</span><span style="margin-left:.25em;">951</span></span></td>
<td>0.9999</td>
<td><span class="nowrap"><span data-sort-value="7000389059188641300♠"></span>3.890<span style="margin-left:.25em;">591</span><span style="margin-left:.25em;">886</span><span style="margin-left:.25em;">413</span></span>
</td></tr>
<tr>
<td>0.95</td>
<td><span class="nowrap"><span data-sort-value="7000195996398454000♠"></span>1.959<span style="margin-left:.25em;">963</span><span style="margin-left:.25em;">984</span><span style="margin-left:.25em;">540</span></span></td>
<td>0.99999</td>
<td><span class="nowrap"><span data-sort-value="7000441717341346900♠"></span>4.417<span style="margin-left:.25em;">173</span><span style="margin-left:.25em;">413</span><span style="margin-left:.25em;">469</span></span>
</td></tr>
<tr>
<td>0.98</td>
<td><span class="nowrap"><span data-sort-value="7000232634787404100♠"></span>2.326<span style="margin-left:.25em;">347</span><span style="margin-left:.25em;">874</span><span style="margin-left:.25em;">041</span></span></td>
<td>0.999999</td>
<td><span class="nowrap"><span data-sort-value="7000489163847569899♠"></span>4.891<span style="margin-left:.25em;">638</span><span style="margin-left:.25em;">475</span><span style="margin-left:.25em;">699</span></span>
</td></tr>
<tr>
<td>0.99</td>
<td><span class="nowrap"><span data-sort-value="7000257582930354900♠"></span>2.575<span style="margin-left:.25em;">829</span><span style="margin-left:.25em;">303</span><span style="margin-left:.25em;">549</span></span></td>
<td>0.9999999</td>
<td><span class="nowrap"><span data-sort-value="7000532672388638400♠"></span>5.326<span style="margin-left:.25em;">723</span><span style="margin-left:.25em;">886</span><span style="margin-left:.25em;">384</span></span>
</td></tr>
<tr>
<td>0.995</td>
<td><span class="nowrap"><span data-sort-value="7000280703376834400♠"></span>2.807<span style="margin-left:.25em;">033</span><span style="margin-left:.25em;">768</span><span style="margin-left:.25em;">344</span></span></td>
<td>0.99999999</td>
<td><span class="nowrap"><span data-sort-value="7000573072886823600♠"></span>5.730<span style="margin-left:.25em;">728</span><span style="margin-left:.25em;">868</span><span style="margin-left:.25em;">236</span></span>
</td></tr>
<tr>
<td>0.998</td>
<td><span class="nowrap"><span data-sort-value="7000309023230616800♠"></span>3.090<span style="margin-left:.25em;">232</span><span style="margin-left:.25em;">306</span><span style="margin-left:.25em;">168</span></span></td>
<td>0.999999999</td>
<td><span class="nowrap"><span data-sort-value="7000610941020486900♠"></span>6.109<span style="margin-left:.25em;">410</span><span style="margin-left:.25em;">204</span><span style="margin-left:.25em;">869</span></span>
</td></tr></tbody></table>
<p>For small <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="p"></span>, the quantile function has the useful <a href="/wiki/Asymptotic_expansion" title="Asymptotic expansion">asymptotic expansion</a>
<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi ^{-1}(p)=-{\sqrt {\ln {\frac {1}{p^{2}}}-\ln \ln {\frac {1}{p^{2}}}-\ln(2\pi )}}+{\mathcal {o}}(1).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>p</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>ln</mi>
<mo>&#x2061;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<msup>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mfrac>
</mrow>
<mo>&#x2212;<!-- --></mo>
<mi>ln</mi>
<mo>&#x2061;<!-- --></mo>
<mi>ln</mi>
<mo>&#x2061;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<msup>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mfrac>
</mrow>
<mo>&#x2212;<!-- --></mo>
<mi>ln</mi>
<mo>&#x2061;<!-- --></mo>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mi>&#x03C0;<!-- π --></mi>
<mo stretchy="false">)</mo>
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<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">o</mi>
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<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Phi ^{-1}(p)=-{\sqrt {\ln {\frac {1}{p^{2}}}-\ln \ln {\frac {1}{p^{2}}}-\ln(2\pi )}}+{\mathcal {o}}(1).}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e58eaaf32eb1ff233e2cfa5649e59a421be6b5fc" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:46.829ex; height:7.509ex;" alt="{\displaystyle \Phi ^{-1}(p)=-{\sqrt {\ln {\frac {1}{p^{2}}}-\ln \ln {\frac {1}{p^{2}}}-\ln(2\pi )}}+{\mathcal {o}}(1).}"></span><sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">&#91;<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (February 2023)">citation needed</span></a></i>&#93;</sup>
</p>
<h2><span class="mw-headline" id="Properties">Properties</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=12" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h2>
<p>The normal distribution is the only distribution whose <a href="/wiki/Cumulant" title="Cumulant">cumulants</a> beyond the first two (i.e., other than the mean and <a href="/wiki/Variance" title="Variance">variance</a>) are zero. It is also the continuous distribution with the <a href="/wiki/Maximum_entropy_probability_distribution" title="Maximum entropy probability distribution">maximum entropy</a> for a specified mean and variance.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15">&#91;15&#93;</a></sup><sup id="cite_ref-16" class="reference"><a href="#cite_note-16">&#91;16&#93;</a></sup> Geary has shown, assuming that the mean and variance are finite, that the normal distribution is the only distribution where the mean and variance calculated from a set of independent draws are independent of each other.<sup id="cite_ref-Geary1936_17-0" class="reference"><a href="#cite_note-Geary1936-17">&#91;17&#93;</a></sup><sup id="cite_ref-18" class="reference"><a href="#cite_note-18">&#91;18&#93;</a></sup>
</p><p>The normal distribution is a subclass of the <a href="/wiki/Elliptical_distribution" title="Elliptical distribution">elliptical distributions</a>. The normal distribution is <a href="/wiki/Symmetric_distribution" class="mw-redirect" title="Symmetric distribution">symmetric</a> about its mean, and is non-zero over the entire real line. As such it may not be a suitable model for variables that are inherently positive or strongly skewed, such as the <a href="/wiki/Weight" title="Weight">weight</a> of a person or the price of a <a href="/wiki/Share_(finance)" title="Share (finance)">share</a>. Such variables may be better described by other distributions, such as the <a href="/wiki/Log-normal_distribution" title="Log-normal distribution">log-normal distribution</a> or the <a href="/wiki/Pareto_distribution" title="Pareto distribution">Pareto distribution</a>.
</p><p>The value of the normal distribution is practically zero when the value <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="x"></span> lies more than a few <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviations</a> away from the mean (e.g., a spread of three standard deviations covers all but 0.27% of the total distribution). Therefore, it may not be an appropriate model when one expects a significant fraction of <a href="/wiki/Outlier" title="Outlier">outliers</a>—values that lie many standard deviations away from the mean—and least squares and other <a href="/wiki/Statistical_inference" title="Statistical inference">statistical inference</a> methods that are optimal for normally distributed variables often become highly unreliable when applied to such data. In those cases, a more <a href="/wiki/Heavy-tailed" class="mw-redirect" title="Heavy-tailed">heavy-tailed</a> distribution should be assumed and the appropriate <a href="/wiki/Robust_statistics" title="Robust statistics">robust statistical inference</a> methods applied.
</p><p>The Gaussian distribution belongs to the family of <a href="/wiki/Stable_distribution" title="Stable distribution">stable distributions</a> which are the attractors of sums of <a href="/wiki/Independent_and_identically_distributed_random_variables" title="Independent and identically distributed random variables">independent, identically distributed</a> distributions whether or not the mean or variance is finite. Except for the Gaussian which is a limiting case, all stable distributions have heavy tails and infinite variance. It is one of the few distributions that are stable and that have probability density functions that can be expressed analytically, the others being the <a href="/wiki/Cauchy_distribution" title="Cauchy distribution">Cauchy distribution</a> and the <a href="/wiki/L%C3%A9vy_distribution" title="Lévy distribution">Lévy distribution</a>.
</p>
<h3><span class="mw-headline" id="Symmetries_and_derivatives">Symmetries and derivatives</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=13" title="Edit section: Symmetries and derivatives"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
<p>The normal distribution with density <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="f(x)"></span> (mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="\mu "></span> and standard deviation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma &gt;0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03C3;<!-- σ --></mi>
<mo>&gt;</mo>
<mn>0</mn>
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<annotation encoding="application/x-tex">{\displaystyle \sigma &gt;0}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/762ecd0f0905dd0d4d7a07f80fa8bfb324b9b021" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="\sigma &gt;0"></span>) has the following properties:
</p>
<ul><li>It is symmetric around the point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\mu ,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo>=</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>,</mo>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle x=\mu ,}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00ba4f6bc9badf6acd7e8d5aef080b99c8ae1601" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.477ex; height:2.176ex;" alt="{\displaystyle x=\mu ,}"></span> which is at the same time the <a href="/wiki/Mode_(statistics)" title="Mode (statistics)">mode</a>, the <a href="/wiki/Median" title="Median">median</a> and the <a href="/wiki/Mean" title="Mean">mean</a> of the distribution.<sup id="cite_ref-PR2.1.4_19-0" class="reference"><a href="#cite_note-PR2.1.4-19">&#91;19&#93;</a></sup></li>
<li>It is <a href="/wiki/Unimodal" class="mw-redirect" title="Unimodal">unimodal</a>: its first <a href="/wiki/Derivative" title="Derivative">derivative</a> is positive for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x&lt;\mu ,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo>&lt;</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>,</mo>
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<annotation encoding="application/x-tex">{\displaystyle x&lt;\mu ,}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4736eab43593e1ad7849b14ddaef32e08b0ffbc" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.477ex; height:2.343ex;" alt="{\displaystyle x&lt;\mu ,}"></span> negative for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x&gt;\mu ,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo>&gt;</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>,</mo>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle x&gt;\mu ,}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3463a22382792e476326893bbc79d8ca5ed60c3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.477ex; height:2.343ex;" alt="{\displaystyle x&gt;\mu ,}"></span> and zero only at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\mu .}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo>=</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>.</mo>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle x=\mu .}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4e68eaf87e1531046ed1ce647c0bcca4d374d1c" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.477ex; height:2.176ex;" alt="{\displaystyle x=\mu .}"></span></li>
<li>The area bounded by the curve and the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="x"></span>-axis is unity (i.e. equal to one).</li>
<li>Its first derivative is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'(x)=-{\frac {x-\mu }{\sigma ^{2}}}f(x).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>f</mi>
<mo>&#x2032;</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>x</mi>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
</mrow>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mfrac>
</mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f'(x)=-{\frac {x-\mu }{\sigma ^{2}}}f(x).}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/889a541adaf23c51ee6b3904fa2419cb1a9a1c66" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:21.523ex; height:5.509ex;" alt="{\displaystyle f&#039;(x)=-{\frac {x-\mu }{\sigma ^{2}}}f(x).}"></span></li>
<li>Its second derivative is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f''(x)={\frac {(x-\mu )^{2}-\sigma ^{2}}{\sigma ^{4}}}f(x).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>f</mi>
<mo>&#x2033;</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>&#x2212;<!-- --></mo>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
</mfrac>
</mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f''(x)={\frac {(x-\mu )^{2}-\sigma ^{2}}{\sigma ^{4}}}f(x).}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e5d9049f7d7e24d492bafca023c1b0d72c6559f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:28.256ex; height:6.176ex;" alt="{\displaystyle f&#039;&#039;(x)={\frac {(x-\mu )^{2}-\sigma ^{2}}{\sigma ^{4}}}f(x).}"></span></li>
<li>Its density has two <a href="/wiki/Inflection_point" title="Inflection point">inflection points</a> (where the second derivative of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="f"></span> is zero and changes sign), located one standard deviation away from the mean, namely at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\mu -\sigma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo>=</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03C3;<!-- σ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x=\mu -\sigma }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccf75fff7d29072e1b95674c36f60694295bc195" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10ex; height:2.509ex;" alt="{\displaystyle x=\mu -\sigma }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\mu +\sigma .}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo>=</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>+</mo>
<mi>&#x03C3;<!-- σ --></mi>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x=\mu +\sigma .}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a956916ad35d8f0a29a2e626c39deda957e7f291" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.647ex; height:2.509ex;" alt="{\displaystyle x=\mu +\sigma .}"></span><sup id="cite_ref-PR2.1.4_19-1" class="reference"><a href="#cite_note-PR2.1.4-19">&#91;19&#93;</a></sup></li>
<li>Its density is <a href="/wiki/Logarithmically_concave_function" title="Logarithmically concave function">log-concave</a>.<sup id="cite_ref-PR2.1.4_19-2" class="reference"><a href="#cite_note-PR2.1.4-19">&#91;19&#93;</a></sup></li>
<li>Its density is infinitely <a href="/wiki/Differentiable_function" title="Differentiable function">differentiable</a>, indeed <a href="/wiki/Supersmooth" class="mw-redirect" title="Supersmooth">supersmooth</a> of order 2.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20">&#91;20&#93;</a></sup></li></ul>
<p>Furthermore, the density <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03C6;<!-- φ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="\varphi "></span> of the standard normal distribution (i.e. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu =0}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3753282c0ad2ea1e7d63f39425efd13c37da3169" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.663ex; height:2.676ex;" alt="\mu =0"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma =1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03C3;<!-- σ --></mi>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma =1}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f759e9b01b4c117d116da9f6d0e635b2247ee502" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle \sigma =1}"></span>) also has the following properties:
</p>
<ul><li>Its first derivative is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi '(x)=-x\varphi (x).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>&#x03C6;<!-- φ --></mi>
<mo>&#x2032;</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>&#x2212;<!-- --></mo>
<mi>x</mi>
<mi>&#x03C6;<!-- φ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \varphi '(x)=-x\varphi (x).}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d95cb5a4c3a2bd3bc56c5a42815570aa18322c9b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.886ex; height:3.009ex;" alt="{\displaystyle \varphi &#039;(x)=-x\varphi (x).}"></span></li>
<li>Its second derivative is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ''(x)=(x^{2}-1)\varphi (x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>&#x03C6;<!-- φ --></mi>
<mo>&#x2033;</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo stretchy="false">(</mo>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mi>&#x03C6;<!-- φ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \varphi ''(x)=(x^{2}-1)\varphi (x)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/daef5d384de735a14c24dae7326ed8852252b488" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.75ex; height:3.176ex;" alt="{\displaystyle \varphi &#039;&#039;(x)=(x^{2}-1)\varphi (x)}"></span></li>
<li>More generally, its <span class="texhtml mvar" style="font-style:italic;">n</span>th derivative is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ^{(n)}(x)=(-1)^{n}\operatorname {He} _{n}(x)\varphi (x),}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>&#x03C6;<!-- φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<msub>
<mi>He</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>&#x2061;<!-- --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mi>&#x03C6;<!-- φ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \varphi ^{(n)}(x)=(-1)^{n}\operatorname {He} _{n}(x)\varphi (x),}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2ba13000c0fb8936cf5dae7d671cc01633da524" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.08ex; height:3.343ex;" alt="{\displaystyle \varphi ^{(n)}(x)=(-1)^{n}\operatorname {He} _{n}(x)\varphi (x),}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {He} _{n}(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>He</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>&#x2061;<!-- --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {He} _{n}(x)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a14cb7440e41048805a63f4c154f4077365f2136" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.133ex; height:2.843ex;" alt="{\displaystyle \operatorname {He} _{n}(x)}"></span> is the <span class="texhtml mvar" style="font-style:italic;">n</span>th (probabilist) <a href="/wiki/Hermite_polynomial" class="mw-redirect" title="Hermite polynomial">Hermite polynomial</a>.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21">&#91;21&#93;</a></sup></li>
<li>The probability that a normally distributed variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"></span> with known <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="\mu "></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03C3;<!-- σ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="\sigma "></span> is in a particular set, can be calculated by using the fact that the fraction <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z=(X-\mu )/\sigma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Z</mi>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>&#x03C3;<!-- σ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Z=(X-\mu )/\sigma }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25037543a35e0e5689bd9bf285d59e083e5382aa" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.302ex; height:2.843ex;" alt="{\displaystyle Z=(X-\mu )/\sigma }"></span> has a standard normal distribution.</li></ul>
<h3><span class="mw-headline" id="Moments">Moments</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=14" title="Edit section: Moments"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1033289096"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/List_of_integrals_of_Gaussian_functions" title="List of integrals of Gaussian functions">List of integrals of Gaussian functions</a></div>
<p>The plain and absolute <a href="/wiki/Moment_(mathematics)" title="Moment (mathematics)">moments</a> of a variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"></span> are the expected values of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X^{p}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X^{p}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34f913b56a87739c3ae2f5f97211f41f41f46bd1" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.056ex; height:2.343ex;" alt="{\displaystyle X^{p}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |X|^{p}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>X</mi>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>p</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |X|^{p}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b2c42f6ecbb2c09979689d81d2b8ed7a4dcd722" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.333ex; height:3.009ex;" alt="{\displaystyle |X|^{p}}"></span>, respectively. If the expected value <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="\mu "></span> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"></span> is zero, these parameters are called <i>central moments;</i> otherwise, these parameters are called <i>non-central moments.</i> Usually we are interested only in moments with integer order <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mtext>&#xA0;</mtext>
<mi>p</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \ p}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f385a789c147f05d215d99fecd7ff19e8fd40b05" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.75ex; height:2.009ex;" alt="\ p"></span>.
</p><p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"></span> has a normal distribution, the non-central moments exist and are finite for any <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="p"></span> whose real part is greater than&#160;1. For any non-negative integer <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="p"></span>, the plain central moments are:<sup id="cite_ref-22" class="reference"><a href="#cite_note-22">&#91;22&#93;</a></sup>
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} \left[(X-\mu )^{p}\right]={\begin{cases}0&amp;{\text{if }}p{\text{ is odd,}}\\\sigma ^{p}(p-1)!!&amp;{\text{if }}p{\text{ is even.}}\end{cases}}}">
<semantics>
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<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} \left[(X-\mu )^{p}\right]={\begin{cases}0&amp;{\text{if }}p{\text{ is odd,}}\\\sigma ^{p}(p-1)!!&amp;{\text{if }}p{\text{ is even.}}\end{cases}}}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1d2c92b62ac2bbe07a8e475faac29c8cc5f7755" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:41.612ex; height:6.176ex;" alt="{\displaystyle \operatorname {E} \left[(X-\mu )^{p}\right]={\begin{cases}0&amp;{\text{if }}p{\text{ is odd,}}\\\sigma ^{p}(p-1)!!&amp;{\text{if }}p{\text{ is even.}}\end{cases}}}"></span></dd></dl>
<p>Here <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!!}">
<semantics>
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<mi>n</mi>
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<annotation encoding="application/x-tex">{\displaystyle n!!}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/511717d541dba5357928e8d8631f1b4d4f8d5b31" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.688ex; height:2.176ex;" alt="n!!"></span> denotes the <a href="/wiki/Double_factorial" title="Double factorial">double factorial</a>, that is, the product of all numbers from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
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<mi>n</mi>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="n"></span> to&#160;1 that have the same parity as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n.}">
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e59df02a9f67a5da3c220f1244c99a46cc4eb1c6" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.042ex; height:1.676ex;" alt="n."></span>
</p><p>The central absolute moments coincide with plain moments for all even orders, but are nonzero for odd orders. For any non-negative integer <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p,}">
<semantics>
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<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
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<annotation encoding="application/x-tex">{\displaystyle p,}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/393fcf18074cb42eafb26b76c515a1e93e17512c" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.906ex; height:2.009ex;" alt="p,"></span>
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\operatorname {E} \left[|X-\mu |^{p}\right]&amp;=\sigma ^{p}(p-1)!!\cdot {\begin{cases}{\sqrt {\frac {2}{\pi }}}&amp;{\text{if }}p{\text{ is odd}}\\1&amp;{\text{if }}p{\text{ is even}}\end{cases}}\\&amp;=\sigma ^{p}\cdot {\frac {2^{p/2}\Gamma \left({\frac {p+1}{2}}\right)}{\sqrt {\pi }}}.\end{aligned}}}">
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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {E} \left[|X-\mu |^{p}\right]&amp;=\sigma ^{p}(p-1)!!\cdot {\begin{cases}{\sqrt {\frac {2}{\pi }}}&amp;{\text{if }}p{\text{ is odd}}\\1&amp;{\text{if }}p{\text{ is even}}\end{cases}}\\&amp;=\sigma ^{p}\cdot {\frac {2^{p/2}\Gamma \left({\frac {p+1}{2}}\right)}{\sqrt {\pi }}}.\end{aligned}}}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b196371c491676efa7ea7770ef56773db7652cd" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -7.505ex; width:47.112ex; height:16.176ex;" alt="{\displaystyle {\begin{aligned}\operatorname {E} \left[|X-\mu |^{p}\right]&amp;=\sigma ^{p}(p-1)!!\cdot {\begin{cases}{\sqrt {\frac {2}{\pi }}}&amp;{\text{if }}p{\text{ is odd}}\\1&amp;{\text{if }}p{\text{ is even}}\end{cases}}\\&amp;=\sigma ^{p}\cdot {\frac {2^{p/2}\Gamma \left({\frac {p+1}{2}}\right)}{\sqrt {\pi }}}.\end{aligned}}}"></span></dd></dl>
<p>The last formula is valid also for any non-integer <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p&gt;-1.}">
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<annotation encoding="application/x-tex">{\displaystyle p&gt;-1.}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2664add830e3fdc570b1dddcdbe85950c3055332" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:7.975ex; height:2.509ex;" alt="{\displaystyle p&gt;-1.}"></span> When the mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \neq 0,}">
<semantics>
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<annotation encoding="application/x-tex">{\displaystyle \mu \neq 0,}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76af80092e410e9ab5ed84403ce73aa79c472ba3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.309ex; height:2.676ex;" alt="{\displaystyle \mu \neq 0,}"></span> the plain and absolute moments can be expressed in terms of <a href="/wiki/Confluent_hypergeometric_function" title="Confluent hypergeometric function">confluent hypergeometric functions</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}_{1}F_{1}}">
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<msub>
<mrow class="MJX-TeXAtom-ORD">
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<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
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<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
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<annotation encoding="application/x-tex">{\displaystyle {}_{1}F_{1}}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e2502464bbbb15d490d62764c2978db65d64d06" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.603ex; height:2.509ex;" alt="{}_{1}F_{1}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U.}">
<semantics>
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<mstyle displaystyle="true" scriptlevel="0">
<mi>U</mi>
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<annotation encoding="application/x-tex">{\displaystyle U.}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a305ef479ab152035f334467a2c314baa23eb36" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.429ex; height:2.176ex;" alt="U."></span><sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">&#91;<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (June 2010)">citation needed</span></a></i>&#93;</sup>
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\operatorname {E} \left[X^{p}\right]&amp;=\sigma ^{p}\cdot (-i{\sqrt {2}})^{p}U\left(-{\frac {p}{2}},{\frac {1}{2}},-{\frac {1}{2}}\left({\frac {\mu }{\sigma }}\right)^{2}\right),\\\operatorname {E} \left[|X|^{p}\right]&amp;=\sigma ^{p}\cdot 2^{p/2}{\frac {\Gamma \left({\frac {1+p}{2}}\right)}{\sqrt {\pi }}}{}_{1}F_{1}\left(-{\frac {p}{2}},{\frac {1}{2}},-{\frac {1}{2}}\left({\frac {\mu }{\sigma }}\right)^{2}\right).\end{aligned}}}">
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<mo>(</mo>
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<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>)</mo>
</mrow>
<mo>.</mo>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {E} \left[X^{p}\right]&amp;=\sigma ^{p}\cdot (-i{\sqrt {2}})^{p}U\left(-{\frac {p}{2}},{\frac {1}{2}},-{\frac {1}{2}}\left({\frac {\mu }{\sigma }}\right)^{2}\right),\\\operatorname {E} \left[|X|^{p}\right]&amp;=\sigma ^{p}\cdot 2^{p/2}{\frac {\Gamma \left({\frac {1+p}{2}}\right)}{\sqrt {\pi }}}{}_{1}F_{1}\left(-{\frac {p}{2}},{\frac {1}{2}},-{\frac {1}{2}}\left({\frac {\mu }{\sigma }}\right)^{2}\right).\end{aligned}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c17bf881593b86e728bf5dfbdb41a4b86da3875" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:54.408ex; height:14.843ex;" alt="{\displaystyle {\begin{aligned}\operatorname {E} \left[X^{p}\right]&amp;=\sigma ^{p}\cdot (-i{\sqrt {2}})^{p}U\left(-{\frac {p}{2}},{\frac {1}{2}},-{\frac {1}{2}}\left({\frac {\mu }{\sigma }}\right)^{2}\right),\\\operatorname {E} \left[|X|^{p}\right]&amp;=\sigma ^{p}\cdot 2^{p/2}{\frac {\Gamma \left({\frac {1+p}{2}}\right)}{\sqrt {\pi }}}{}_{1}F_{1}\left(-{\frac {p}{2}},{\frac {1}{2}},-{\frac {1}{2}}\left({\frac {\mu }{\sigma }}\right)^{2}\right).\end{aligned}}}"></span></dd></dl>
<p>These expressions remain valid even if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="p"></span> is not an integer. See also <a href="/wiki/Hermite_polynomials#&quot;Negative_variance&quot;" title="Hermite polynomials">generalized Hermite polynomials</a>.
</p>
<table class="wikitable" style="background:#fff; margin: auto;">
<tbody><tr>
<th>Order</th>
<th>Non-central moment</th>
<th>Central moment
</th></tr>
<tr>
<td>1
</td>
<td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="\mu "></span>
</td>
<td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span>
</td></tr>
<tr>
<td>2
</td>
<td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu ^{2}+\sigma ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>+</mo>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu ^{2}+\sigma ^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49112e161897039a88a162ff2ad10ea4a8c9e8ac" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.681ex; height:3.176ex;" alt="{\displaystyle \mu ^{2}+\sigma ^{2}}"></span>
</td>
<td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="\sigma ^{2}"></span>
</td></tr>
<tr>
<td>3
</td>
<td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu ^{3}+3\mu \sigma ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
<mo>+</mo>
<mn>3</mn>
<mi>&#x03BC;<!-- μ --></mi>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu ^{3}+3\mu \sigma ^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/036df43f1879ad3e1e36a3394d3428475217891c" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.245ex; height:3.176ex;" alt="{\displaystyle \mu ^{3}+3\mu \sigma ^{2}}"></span>
</td>
<td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span>
</td></tr>
<tr>
<td>4
</td>
<td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu ^{4}+6\mu ^{2}\sigma ^{2}+3\sigma ^{4}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
<mo>+</mo>
<mn>6</mn>
<msup>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>+</mo>
<mn>3</mn>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu ^{4}+6\mu ^{2}\sigma ^{2}+3\sigma ^{4}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a0c44283f14f944c968ea3c5c9fd20cd905a7eb" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.687ex; height:3.176ex;" alt="{\displaystyle \mu ^{4}+6\mu ^{2}\sigma ^{2}+3\sigma ^{4}}"></span>
</td>
<td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\sigma ^{4}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>3</mn>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 3\sigma ^{4}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60711695348caa7b632267f93d69be5c1e70993e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.547ex; height:2.676ex;" alt="{\displaystyle 3\sigma ^{4}}"></span>
</td></tr>
<tr>
<td>5
</td>
<td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu ^{5}+10\mu ^{3}\sigma ^{2}+15\mu \sigma ^{4}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>5</mn>
</mrow>
</msup>
<mo>+</mo>
<mn>10</mn>
<msup>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>+</mo>
<mn>15</mn>
<mi>&#x03BC;<!-- μ --></mi>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu ^{5}+10\mu ^{3}\sigma ^{2}+15\mu \sigma ^{4}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f3c84e4530f6c884882907fba25c498f189e19f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.414ex; height:3.176ex;" alt="{\displaystyle \mu ^{5}+10\mu ^{3}\sigma ^{2}+15\mu \sigma ^{4}}"></span>
</td>
<td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span>
</td></tr>
<tr>
<td>6
</td>
<td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu ^{6}+15\mu ^{4}\sigma ^{2}+45\mu ^{2}\sigma ^{4}+15\sigma ^{6}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>6</mn>
</mrow>
</msup>
<mo>+</mo>
<mn>15</mn>
<msup>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>+</mo>
<mn>45</mn>
<msup>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
<mo>+</mo>
<mn>15</mn>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>6</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu ^{6}+15\mu ^{4}\sigma ^{2}+45\mu ^{2}\sigma ^{4}+15\sigma ^{6}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/beda55a9d202328e1f0836b703aa646cf21b1ee8" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.018ex; height:3.176ex;" alt="{\displaystyle \mu ^{6}+15\mu ^{4}\sigma ^{2}+45\mu ^{2}\sigma ^{4}+15\sigma ^{6}}"></span>
</td>
<td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 15\sigma ^{6}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>15</mn>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>6</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 15\sigma ^{6}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2ecc5219c1a127d5dd5b79987ea76eba8b5c59a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.71ex; height:2.676ex;" alt="{\displaystyle 15\sigma ^{6}}"></span>
</td></tr>
<tr>
<td>7
</td>
<td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu ^{7}+21\mu ^{5}\sigma ^{2}+105\mu ^{3}\sigma ^{4}+105\mu \sigma ^{6}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>7</mn>
</mrow>
</msup>
<mo>+</mo>
<mn>21</mn>
<msup>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>5</mn>
</mrow>
</msup>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>+</mo>
<mn>105</mn>
<msup>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
<mo>+</mo>
<mn>105</mn>
<mi>&#x03BC;<!-- μ --></mi>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>6</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu ^{7}+21\mu ^{5}\sigma ^{2}+105\mu ^{3}\sigma ^{4}+105\mu \sigma ^{6}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db8ed0b46ce6cef2af54de84e6801c3a1c26328a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.744ex; height:3.176ex;" alt="{\displaystyle \mu ^{7}+21\mu ^{5}\sigma ^{2}+105\mu ^{3}\sigma ^{4}+105\mu \sigma ^{6}}"></span>
</td>
<td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span>
</td></tr>
<tr>
<td>8
</td>
<td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu ^{8}+28\mu ^{6}\sigma ^{2}+210\mu ^{4}\sigma ^{4}+420\mu ^{2}\sigma ^{6}+105\sigma ^{8}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>8</mn>
</mrow>
</msup>
<mo>+</mo>
<mn>28</mn>
<msup>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>6</mn>
</mrow>
</msup>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>+</mo>
<mn>210</mn>
<msup>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
<mo>+</mo>
<mn>420</mn>
<msup>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>6</mn>
</mrow>
</msup>
<mo>+</mo>
<mn>105</mn>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>8</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu ^{8}+28\mu ^{6}\sigma ^{2}+210\mu ^{4}\sigma ^{4}+420\mu ^{2}\sigma ^{6}+105\sigma ^{8}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff3470c8027342e4381868a630ab1580c92fff4a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.511ex; height:3.176ex;" alt="{\displaystyle \mu ^{8}+28\mu ^{6}\sigma ^{2}+210\mu ^{4}\sigma ^{4}+420\mu ^{2}\sigma ^{6}+105\sigma ^{8}}"></span>
</td>
<td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 105\sigma ^{8}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>105</mn>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>8</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 105\sigma ^{8}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7de4246264d28e26590465fac40c83e06bfbef4" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.872ex; height:2.676ex;" alt="{\displaystyle 105\sigma ^{8}}"></span>
</td></tr></tbody></table>
<p>The expectation of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"></span> conditioned on the event that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"></span> lies in an interval <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<mi>a</mi>
<mo>,</mo>
<mi>b</mi>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="[a,b]"></span> is given by
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} \left[X\mid a&lt;X&lt;b\right]=\mu -\sigma ^{2}{\frac {f(b)-f(a)}{F(b)-F(a)}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">E</mi>
<mo>&#x2061;<!-- --></mo>
<mrow>
<mo>[</mo>
<mrow>
<mi>X</mi>
<mo>&#x2223;<!-- --></mo>
<mi>a</mi>
<mo>&lt;</mo>
<mi>X</mi>
<mo>&lt;</mo>
<mi>b</mi>
</mrow>
<mo>]</mo>
</mrow>
<mo>=</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>&#x2212;<!-- --></mo>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo stretchy="false">)</mo>
<mo>&#x2212;<!-- --></mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo stretchy="false">)</mo>
<mo>&#x2212;<!-- --></mo>
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} \left[X\mid a&lt;X&lt;b\right]=\mu -\sigma ^{2}{\frac {f(b)-f(a)}{F(b)-F(a)}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d82ec10bf31f0b63137699ae6e2b5a346770b097" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:39.927ex; height:6.509ex;" alt="{\displaystyle \operatorname {E} \left[X\mid a&lt;X&lt;b\right]=\mu -\sigma ^{2}{\frac {f(b)-f(a)}{F(b)-F(a)}}}"></span></dd></dl>
<p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="f"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="F"></span> respectively are the density and the cumulative distribution function of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"></span>. For <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=\infty }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>b</mi>
<mo>=</mo>
<mi mathvariant="normal">&#x221E;<!-- ∞ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b=\infty }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e4fa901adf7dab615335dc0ceb57480451a70ec" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.42ex; height:2.176ex;" alt="b=\infty "></span> this is known as the <a href="/wiki/Inverse_Mills_ratio" class="mw-redirect" title="Inverse Mills ratio">inverse Mills ratio</a>. Note that above, density <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="f"></span> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"></span> is used instead of standard normal density as in inverse Mills ratio, so here we have <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="\sigma ^{2}"></span> instead of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03C3;<!-- σ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="\sigma "></span>.
</p>
<h3><span class="mw-headline" id="Fourier_transform_and_characteristic_function">Fourier transform and characteristic function</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=15" title="Edit section: Fourier transform and characteristic function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
<p>The <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a> of a normal density <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="f"></span> with mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="\mu "></span> and standard deviation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03C3;<!-- σ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="\sigma "></span> is<sup id="cite_ref-23" class="reference"><a href="#cite_note-23">&#91;23&#93;</a></sup>
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f}}(t)=\int _{-\infty }^{\infty }f(x)e^{-itx}\,dx=e^{-i\mu t}e^{-{\frac {1}{2}}(\sigma t)^{2}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>f</mi>
<mo stretchy="false">&#x005E;<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msubsup>
<mo>&#x222B;<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<mi mathvariant="normal">&#x221E;<!-- ∞ --></mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">&#x221E;<!-- ∞ --></mi>
</mrow>
</msubsup>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<mi>i</mi>
<mi>t</mi>
<mi>x</mi>
</mrow>
</msup>
<mspace width="thinmathspace" />
<mi>d</mi>
<mi>x</mi>
<mo>=</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<mi>i</mi>
<mi>&#x03BC;<!-- μ --></mi>
<mi>t</mi>
</mrow>
</msup>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo stretchy="false">(</mo>
<mi>&#x03C3;<!-- σ --></mi>
<mi>t</mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\hat {f}}(t)=\int _{-\infty }^{\infty }f(x)e^{-itx}\,dx=e^{-i\mu t}e^{-{\frac {1}{2}}(\sigma t)^{2}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/776d1d19793475151305e22947b74646d47bfc91" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.913ex; height:6.009ex;" alt="{\displaystyle {\hat {f}}(t)=\int _{-\infty }^{\infty }f(x)e^{-itx}\,dx=e^{-i\mu t}e^{-{\frac {1}{2}}(\sigma t)^{2}}}"></span></dd></dl>
<p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="i"></span> is the <a href="/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a>. If the mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu =0}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3753282c0ad2ea1e7d63f39425efd13c37da3169" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.663ex; height:2.676ex;" alt="\mu =0"></span>, the first factor is 1, and the Fourier transform is, apart from a constant factor, a normal density on the <a href="/wiki/Frequency_domain" title="Frequency domain">frequency domain</a>, with mean 0 and standard deviation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/\sigma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>&#x03C3;<!-- σ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 1/\sigma }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00d5187486468042e9692b18c216b60679aafef3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.655ex; height:2.843ex;" alt="1/\sigma "></span>. In particular, the standard normal distribution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03C6;<!-- φ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="\varphi "></span> is an <a href="/wiki/Fourier_transform#Eigenfunctions" title="Fourier transform">eigenfunction</a> of the Fourier transform.
</p><p>In probability theory, the Fourier transform of the probability distribution of a real-valued random variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"></span> is closely connected to the <a href="/wiki/Characteristic_function_(probability_theory)" title="Characteristic function (probability theory)">characteristic function</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{X}(t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>&#x03C6;<!-- φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
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</msub>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \varphi _{X}(t)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2578322e22b80aa79b1e5a4aebf144e5d642c8a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.801ex; height:2.843ex;" alt="\varphi _{X}(t)"></span> of that variable, which is defined as the <a href="/wiki/Expected_value" title="Expected value">expected value</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{itX}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>t</mi>
<mi>X</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e^{itX}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6835fb2b6b1ef511c0bd711e67dd256360e9dd39" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.877ex; height:2.676ex;" alt="e^{{itX}}"></span>, as a function of the real variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle t}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="t"></span> (the <a href="/wiki/Frequency" title="Frequency">frequency</a> parameter of the Fourier transform). This definition can be analytically extended to a complex-value variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle t}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="t"></span>.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24">&#91;24&#93;</a></sup> The relation between both is:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{X}(t)={\hat {f}}(-t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>&#x03C6;<!-- φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>f</mi>
<mo stretchy="false">&#x005E;<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mo>&#x2212;<!-- --></mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \varphi _{X}(t)={\hat {f}}(-t)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b65e000bb190ed0337f893095d5d72d2dbd2bcfe" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.056ex; height:3.343ex;" alt="{\displaystyle \varphi _{X}(t)={\hat {f}}(-t)}"></span></dd></dl>
<h3><span class="mw-headline" id="Moment-_and_cumulant-generating_functions">Moment- and cumulant-generating functions</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=16" title="Edit section: Moment- and cumulant-generating functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
<p>The <a href="/wiki/Moment_generating_function" class="mw-redirect" title="Moment generating function">moment generating function</a> of a real random variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"></span> is the expected value of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{tX}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
<mi>X</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e^{tX}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b20e8a2103c26e6e2602c9ec39c1acdb7a639ad9" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.31ex; height:2.676ex;" alt="{\displaystyle e^{tX}}"></span>, as a function of the real parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle t}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="t"></span>. For a normal distribution with density <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="f"></span>, mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="\mu "></span> and deviation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03C3;<!-- σ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="\sigma "></span>, the moment generating function exists and is equal to
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M(t)=\operatorname {E} \left[e^{tX}\right]={\hat {f}}(it)=e^{\mu t}e^{\sigma ^{2}t^{2}/2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>M</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi mathvariant="normal">E</mi>
<mo>&#x2061;<!-- --></mo>
<mrow>
<mo>[</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
<mi>X</mi>
</mrow>
</msup>
<mo>]</mo>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>f</mi>
<mo stretchy="false">&#x005E;<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mi>i</mi>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>&#x03BC;<!-- μ --></mi>
<mi>t</mi>
</mrow>
</msup>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle M(t)=\operatorname {E} \left[e^{tX}\right]={\hat {f}}(it)=e^{\mu t}e^{\sigma ^{2}t^{2}/2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/935bdfb329038ee45bf7cc94d83f68b66a5c74c5" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.427ex; height:3.676ex;" alt="{\displaystyle M(t)=\operatorname {E} \left[e^{tX}\right]={\hat {f}}(it)=e^{\mu t}e^{\sigma ^{2}t^{2}/2}}"></span></dd></dl>
<p>The <a href="/wiki/Cumulant_generating_function" class="mw-redirect" title="Cumulant generating function">cumulant generating function</a> is the logarithm of the moment generating function, namely
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(t)=\ln M(t)=\mu t+{\tfrac {1}{2}}\sigma ^{2}t^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>ln</mi>
<mo>&#x2061;<!-- --></mo>
<mi>M</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mi>t</mi>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
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</mrow>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g(t)=\ln M(t)=\mu t+{\tfrac {1}{2}}\sigma ^{2}t^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/457daa5e2687fecf756404b31202b8f8cd964436" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:28.398ex; height:3.509ex;" alt="{\displaystyle g(t)=\ln M(t)=\mu t+{\tfrac {1}{2}}\sigma ^{2}t^{2}}"></span></dd></dl>
<p>Since this is a quadratic polynomial in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle t}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="t"></span>, only the first two <a href="/wiki/Cumulant" title="Cumulant">cumulants</a> are nonzero, namely the mean&#160;<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="\mu "></span> and the variance&#160;<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="\sigma ^{2}"></span>.
</p>
<h3><span class="mw-headline" id="Stein_operator_and_class">Stein operator and class</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=17" title="Edit section: Stein operator and class"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
<p>Within <a href="/wiki/Stein%27s_method" title="Stein&#39;s method">Stein's method</a> the Stein operator and class of a random variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim {\mathcal {N}}(\mu ,\sigma ^{2})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>&#x223C;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">N</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>,</mo>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X\sim {\mathcal {N}}(\mu ,\sigma ^{2})}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e49a012c102388008a926ef3e2e28d099d539751" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.983ex; height:3.176ex;" alt="X\sim {\mathcal {N}}(\mu ,\sigma ^{2})"></span> are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {A}}f(x)=\sigma ^{2}f'(x)-(x-\mu )f(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">A</mi>
</mrow>
</mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<msup>
<mi>f</mi>
<mo>&#x2032;</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>&#x2212;<!-- --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo stretchy="false">)</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {A}}f(x)=\sigma ^{2}f'(x)-(x-\mu )f(x)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd542c9a4e963863683da5cb4e52ad898fddf340" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.587ex; height:3.176ex;" alt="{\displaystyle {\mathcal {A}}f(x)=\sigma ^{2}f&#039;(x)-(x-\mu )f(x)}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">F</mi>
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</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\mathcal {F}}"></span> the class of all absolutely continuous functions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {R} \to \mathbb {R} {\mbox{ such that }}\mathbb {E} [|f'(X)|]&lt;\infty }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<mi mathvariant="double-struck">R</mi>
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<mo stretchy="false">&#x2192;<!-- → --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mtext>&#xA0;such that&#xA0;</mtext>
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<mi mathvariant="double-struck">E</mi>
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<annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} \to \mathbb {R} {\mbox{ such that }}\mathbb {E} [|f'(X)|]&lt;\infty }</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69d73a6b7e591a67eaff64aaf974a8c37584626e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.08ex; height:3.009ex;" alt="{\displaystyle f:\mathbb {R} \to \mathbb {R} {\mbox{ such that }}\mathbb {E} [|f&#039;(X)|]&lt;\infty }"></span>.
</p>
<h3><span class="mw-headline" id="Zero-variance_limit">Zero-variance limit</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=18" title="Edit section: Zero-variance limit"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
<p>In the <a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)">limit</a> when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03C3;<!-- σ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="\sigma "></span> tends to zero, the probability density <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
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<mo stretchy="false">(</mo>
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<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="f(x)"></span> eventually tends to zero at any <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\neq \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle x\neq \mu }</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9163f4996800ddbd7d2ee2b4a55297e485d89feb" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.83ex; height:2.676ex;" alt="{\displaystyle x\neq \mu }"></span>, but grows without limit if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<mi>x</mi>
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<mi>&#x03BC;<!-- μ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle x=\mu }</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b20a0e3d1103b2704e577d15b7319bf0870e5d97" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.83ex; height:2.176ex;" alt="{\displaystyle x=\mu }"></span>, while its integral remains equal to 1. Therefore, the normal distribution cannot be defined as an ordinary <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma =0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle \sigma =0}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1eb4831f1e0ca1ba7d007dc6b973e54787e1a4b4" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="\sigma =0"></span>.
</p><p>However, one can define the normal distribution with zero variance as a <a href="/wiki/Generalized_function" title="Generalized function">generalized function</a>; specifically, as a <a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }">
<semantics>
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<mi>&#x03B4;<!-- δ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \delta }</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321cfa797202b3e1f8620663ff43c4660ea03a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="\delta "></span> translated by the mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">
<semantics>
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<mi>&#x03BC;<!-- μ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="\mu "></span>, that is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\delta (x-\mu ).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<mi>f</mi>
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<mo stretchy="false">(</mo>
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<mo>&#x2212;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo stretchy="false">)</mo>
<mo>.</mo>
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<annotation encoding="application/x-tex">{\displaystyle f(x)=\delta (x-\mu ).}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d92c48579df6b85ff2eb7579a6aacbb50fc7fc1e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.593ex; height:2.843ex;" alt="{\displaystyle f(x)=\delta (x-\mu ).}"></span>
Its cumulative distribution function is then the <a href="/wiki/Heaviside_step_function" title="Heaviside step function">Heaviside step function</a> translated by the mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="\mu "></span>, namely
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)={\begin{cases}0&amp;{\text{if }}x&lt;\mu \\1&amp;{\text{if }}x\geq \mu \end{cases}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle F(x)={\begin{cases}0&amp;{\text{if }}x&lt;\mu \\1&amp;{\text{if }}x\geq \mu \end{cases}}}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90400cbbc8895d9f3c9a62d7502ed0f077c6ee3b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.727ex; height:6.176ex;" alt="{\displaystyle F(x)={\begin{cases}0&amp;{\text{if }}x&lt;\mu \\1&amp;{\text{if }}x\geq \mu \end{cases}}}"></span></dd></dl>
<h3><span class="mw-headline" id="Maximum_entropy">Maximum entropy</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=19" title="Edit section: Maximum entropy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
<p>Of all probability distributions over the reals with a specified finite mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<mi>&#x03BC;<!-- μ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="\mu "></span> and finite variance&#160;<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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<annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="\sigma ^{2}"></span>, the normal distribution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(\mu ,\sigma ^{2})}">
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<mn>2</mn>
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</msup>
<mo stretchy="false">)</mo>
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<annotation encoding="application/x-tex">{\displaystyle N(\mu ,\sigma ^{2})}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cbd76720b12f0428a8bf1460b7a67cf2f5f3817" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.693ex; height:3.176ex;" alt="N(\mu ,\sigma ^{2})"></span> is the one with <a href="/wiki/Maximum_entropy_probability_distribution" title="Maximum entropy probability distribution">maximum entropy</a>.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25">&#91;25&#93;</a></sup> To see this, let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"></span> be a <a href="/wiki/Continuous_random_variable" class="mw-redirect" title="Continuous random variable">continuous random variable</a> with <a href="/wiki/Probability_density_function" title="Probability density function">probability density</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="f(x)"></span>. The entropy of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"></span> is defined as<sup id="cite_ref-26" class="reference"><a href="#cite_note-26">&#91;26&#93;</a></sup><sup id="cite_ref-27" class="reference"><a href="#cite_note-27">&#91;27&#93;</a></sup><sup id="cite_ref-28" class="reference"><a href="#cite_note-28">&#91;28&#93;</a></sup>
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(X)=-\int _{-\infty }^{\infty }f(x)\ln f(x)\,dx}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<mo stretchy="false">(</mo>
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<mo stretchy="false">)</mo>
<mspace width="thinmathspace" />
<mi>d</mi>
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<annotation encoding="application/x-tex">{\displaystyle H(X)=-\int _{-\infty }^{\infty }f(x)\ln f(x)\,dx}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2999d7dfcdb197b47e7793268143e5dcbbbb98d4" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.462ex; height:6.009ex;" alt="{\displaystyle H(X)=-\int _{-\infty }^{\infty }f(x)\ln f(x)\,dx}"></span></dd></dl>
<p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)\log f(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
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<mi>log</mi>
<mo>&#x2061;<!-- --></mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(x)\log f(x)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8d506be484aade6667bfd4790bd40d237caedc0" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.581ex; height:2.843ex;" alt="{\displaystyle f(x)\log f(x)}"></span> is understood to be zero whenever <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(x)=0}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf85883d74b75fe35ca8d3f2b44802df078e4fa1" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.678ex; height:2.843ex;" alt="f(x)=0"></span>. This functional can be maximized, subject to the constraints that the distribution is properly normalized and has a specified mean and variance, by using <a href="/wiki/Variational_calculus" class="mw-redirect" title="Variational calculus">variational calculus</a>. A function with three <a href="/wiki/Lagrange_multipliers" class="mw-redirect" title="Lagrange multipliers">Lagrange multipliers</a> is defined:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=-\int _{-\infty }^{\infty }f(x)\ln f(x)\,dx-\lambda _{0}\left(1-\int _{-\infty }^{\infty }f(x)\,dx\right)-\lambda _{1}\left(\mu -\int _{-\infty }^{\infty }f(x)x\,dx\right)-\lambda _{2}\left(\sigma ^{2}-\int _{-\infty }^{\infty }f(x)(x-\mu )^{2}\,dx\right)\,.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>L</mi>
<mo>=</mo>
<mo>&#x2212;<!-- --></mo>
<msubsup>
<mo>&#x222B;<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<mi mathvariant="normal">&#x221E;<!-- ∞ --></mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">&#x221E;<!-- ∞ --></mi>
</mrow>
</msubsup>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mi>ln</mi>
<mo>&#x2061;<!-- --></mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace" />
<mi>d</mi>
<mi>x</mi>
<mo>&#x2212;<!-- --></mo>
<msub>
<mi>&#x03BB;<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>&#x2212;<!-- --></mo>
<msubsup>
<mo>&#x222B;<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<mi mathvariant="normal">&#x221E;<!-- ∞ --></mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">&#x221E;<!-- ∞ --></mi>
</mrow>
</msubsup>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace" />
<mi>d</mi>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>&#x2212;<!-- --></mo>
<msub>
<mi>&#x03BB;<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<mrow>
<mi>&#x03BC;<!-- μ --></mi>
<mo>&#x2212;<!-- --></mo>
<msubsup>
<mo>&#x222B;<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<mi mathvariant="normal">&#x221E;<!-- ∞ --></mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">&#x221E;<!-- ∞ --></mi>
</mrow>
</msubsup>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mi>x</mi>
<mspace width="thinmathspace" />
<mi>d</mi>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>&#x2212;<!-- --></mo>
<msub>
<mi>&#x03BB;<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<mrow>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>&#x2212;<!-- --></mo>
<msubsup>
<mo>&#x222B;<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<mi mathvariant="normal">&#x221E;<!-- ∞ --></mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">&#x221E;<!-- ∞ --></mi>
</mrow>
</msubsup>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mspace width="thinmathspace" />
<mi>d</mi>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mspace width="thinmathspace" />
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle L=-\int _{-\infty }^{\infty }f(x)\ln f(x)\,dx-\lambda _{0}\left(1-\int _{-\infty }^{\infty }f(x)\,dx\right)-\lambda _{1}\left(\mu -\int _{-\infty }^{\infty }f(x)x\,dx\right)-\lambda _{2}\left(\sigma ^{2}-\int _{-\infty }^{\infty }f(x)(x-\mu )^{2}\,dx\right)\,.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/856af00c1ea9a107166c797aeb451a2978d699c3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:114.573ex; height:6.176ex;" alt="{\displaystyle L=-\int _{-\infty }^{\infty }f(x)\ln f(x)\,dx-\lambda _{0}\left(1-\int _{-\infty }^{\infty }f(x)\,dx\right)-\lambda _{1}\left(\mu -\int _{-\infty }^{\infty }f(x)x\,dx\right)-\lambda _{2}\left(\sigma ^{2}-\int _{-\infty }^{\infty }f(x)(x-\mu )^{2}\,dx\right)\,.}"></span></dd></dl>
<p>At maximum entropy, a small variation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta f(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03B4;<!-- δ --></mi>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \delta f(x)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6443aec0d016c556a8d440074b7bb5c4df23232b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.466ex; height:2.843ex;" alt="{\displaystyle \delta f(x)}"></span> about <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="f(x)"></span> will produce a variation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta L}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03B4;<!-- δ --></mi>
<mi>L</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \delta L}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/891517142b0a3696fc42b514c7fd60b304c2f9a7" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.631ex; height:2.343ex;" alt="\delta L"></span> about <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>L</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle L}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="L"></span> which is equal to 0:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=\delta L=\int _{-\infty }^{\infty }\delta f(x)\left(-\ln f(x)-1+\lambda _{0}+\lambda _{1}x+\lambda _{2}(x-\mu )^{2}\right)\,dx}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>0</mn>
<mo>=</mo>
<mi>&#x03B4;<!-- δ --></mi>
<mi>L</mi>
<mo>=</mo>
<msubsup>
<mo>&#x222B;<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<mi mathvariant="normal">&#x221E;<!-- ∞ --></mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">&#x221E;<!-- ∞ --></mi>
</mrow>
</msubsup>
<mi>&#x03B4;<!-- δ --></mi>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>&#x2212;<!-- --></mo>
<mi>ln</mi>
<mo>&#x2061;<!-- --></mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
<mo>+</mo>
<msub>
<mi>&#x03BB;<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>&#x03BB;<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mi>x</mi>
<mo>+</mo>
<msub>
<mi>&#x03BB;<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>)</mo>
</mrow>
<mspace width="thinmathspace" />
<mi>d</mi>
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 0=\delta L=\int _{-\infty }^{\infty }\delta f(x)\left(-\ln f(x)-1+\lambda _{0}+\lambda _{1}x+\lambda _{2}(x-\mu )^{2}\right)\,dx}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01f85ec3c64296bcf3069d9008f652b9ea9fc897" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:64.584ex; height:6.009ex;" alt="{\displaystyle 0=\delta L=\int _{-\infty }^{\infty }\delta f(x)\left(-\ln f(x)-1+\lambda _{0}+\lambda _{1}x+\lambda _{2}(x-\mu )^{2}\right)\,dx}"></span></dd></dl>
<p>Since this must hold for any small <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta f(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03B4;<!-- δ --></mi>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \delta f(x)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6443aec0d016c556a8d440074b7bb5c4df23232b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.466ex; height:2.843ex;" alt="{\displaystyle \delta f(x)}"></span>, the factor multiplying <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta f(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03B4;<!-- δ --></mi>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \delta f(x)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6443aec0d016c556a8d440074b7bb5c4df23232b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.466ex; height:2.843ex;" alt="{\displaystyle \delta f(x)}"></span> must be zero, and solving for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="f(x)"></span> yields:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\exp \left(-1+\lambda _{0}+\lambda _{1}x+\lambda _{2}(x-\mu )^{2}\right)\,.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>exp</mi>
<mo>&#x2061;<!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
<mo>+</mo>
<msub>
<mi>&#x03BB;<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>&#x03BB;<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mi>x</mi>
<mo>+</mo>
<msub>
<mi>&#x03BB;<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>)</mo>
</mrow>
<mspace width="thinmathspace" />
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(x)=\exp \left(-1+\lambda _{0}+\lambda _{1}x+\lambda _{2}(x-\mu )^{2}\right)\,.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a80908345c8c165230ad4e818f29470b33064d5" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:42.717ex; height:3.343ex;" alt="{\displaystyle f(x)=\exp \left(-1+\lambda _{0}+\lambda _{1}x+\lambda _{2}(x-\mu )^{2}\right)\,.}"></span></dd></dl>
<p>The Lagrange constraints that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="f(x)"></span> is properly normalized and has the specified mean and variance are satisfied if and only if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>&#x03BB;<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lambda _{0}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfa5ad1eb6cdaf3d8dfd77991ee9ce7bdf169184" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.409ex; height:2.509ex;" alt="\lambda _{0}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>&#x03BB;<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lambda _{1}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/571a423bece8f29bcd1b48572f18dd4f6213dce2" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.409ex; height:2.509ex;" alt="\lambda _{1}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>&#x03BB;<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lambda _{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b668a1bd1e8ab9452ca975b7497546e7c1ba187" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.409ex; height:2.509ex;" alt="\lambda _{2}"></span> are chosen so that
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}\,.}">
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<mstyle displaystyle="true" scriptlevel="0">
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<annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}\,.}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ac6c71a4a3df62eeaf7e052e27ce356793102f5" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:24.077ex; height:7.176ex;" alt="{\displaystyle f(x)={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}\,.}"></span></dd></dl>
<p>The entropy of a normal distribution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\sim N(\mu ,\sigma ^{2})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>&#x223C;<!-- --></mo>
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<mo stretchy="false">(</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>,</mo>
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<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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<mo stretchy="false">)</mo>
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<annotation encoding="application/x-tex">{\displaystyle X\sim N(\mu ,\sigma ^{2})}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ac7642fb5117eb47ddd41db3006f20fb7886f01" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.772ex; height:3.176ex;" alt="X\sim N(\mu ,\sigma ^{2})"></span> is equal to
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(X)={\tfrac {1}{2}}(1+\ln 2\sigma ^{2}\pi )\,,}">
<semantics>
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<mo>,</mo>
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<annotation encoding="application/x-tex">{\displaystyle H(X)={\tfrac {1}{2}}(1+\ln 2\sigma ^{2}\pi )\,,}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2448beacbfebb0a9ccc54a4927aaa5dde946e77" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:24.661ex; height:3.509ex;" alt="{\displaystyle H(X)={\tfrac {1}{2}}(1+\ln 2\sigma ^{2}\pi )\,,}"></span></dd></dl>
<p>which is independent of the mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="\mu "></span>.
</p>
<h3><span class="mw-headline" id="Other_properties">Other properties</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=20" title="Edit section: Other properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
<div><ol><li>If the characteristic function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{X}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>&#x03D5;<!-- ϕ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \phi _{X}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/232d398e2daa2cdd61f645631835f3f7876e7231" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.018ex; height:2.509ex;" alt="\phi _{X}"></span> of some random variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"></span> is of the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{X}(t)=\exp Q(t)}">
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<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>&#x03D5;<!-- ϕ --></mi>
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<mo stretchy="false">)</mo>
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<mi>exp</mi>
<mo>&#x2061;<!-- --></mo>
<mi>Q</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \phi _{X}(t)=\exp Q(t)}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7f582800b9c48fe8b6cf7ae837ff33d0a77c7aa" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.192ex; height:2.843ex;" alt="{\displaystyle \phi _{X}(t)=\exp Q(t)}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(t)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Q</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Q(t)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68b7ab35402f0f501cbc361f5309fe64fd678cd0" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.487ex; height:2.843ex;" alt="Q(t)"></span> is a <a href="/wiki/Polynomial" title="Polynomial">polynomial</a>, then the <b>Marcinkiewicz theorem</b> (named after <a href="/wiki/J%C3%B3zef_Marcinkiewicz" title="Józef Marcinkiewicz">Józef Marcinkiewicz</a>) asserts that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Q</mi>
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<annotation encoding="application/x-tex">{\displaystyle Q}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="Q"></span> can be at most a quadratic polynomial, and therefore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"></span> is a normal random variable.<sup id="cite_ref-Bryc_1995_35_29-0" class="reference"><a href="#cite_note-Bryc_1995_35-29">&#91;29&#93;</a></sup> The consequence of this result is that the normal distribution is the only distribution with a finite number (two) of non-zero <a href="/wiki/Cumulant" title="Cumulant">cumulants</a>.</li><li>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
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<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="Y"></span> are <a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">jointly normal</a> and <a href="/wiki/Uncorrelated" class="mw-redirect" title="Uncorrelated">uncorrelated</a>, then they are <a href="/wiki/Independence_(probability_theory)" title="Independence (probability theory)">independent</a>. The requirement that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
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<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="Y"></span> should be <i>jointly</i> normal is essential; without it the property does not hold.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30">&#91;30&#93;</a></sup><sup id="cite_ref-31" class="reference"><a href="#cite_note-31">&#91;31&#93;</a></sup><sup><a href="/wiki/Normally_distributed_and_uncorrelated_does_not_imply_independent" title="Normally distributed and uncorrelated does not imply independent">[proof]</a></sup> For non-normal random variables uncorrelatedness does not imply independence.</li><li>The <a href="/wiki/Kullback%E2%80%93Leibler_divergence" title="KullbackLeibler divergence">KullbackLeibler divergence</a> of one normal distribution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}\sim N(\mu _{1},\sigma _{1}^{2})}">
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<mi>X</mi>
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<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle X_{1}\sim N(\mu _{1},\sigma _{1}^{2})}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41a8ceeed9525ae5c45c6622169ae59ea8a9d05e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.824ex; height:3.176ex;" alt="{\displaystyle X_{1}\sim N(\mu _{1},\sigma _{1}^{2})}"></span> from another <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{2}\sim N(\mu _{2},\sigma _{2}^{2})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>&#x223C;<!-- --></mo>
<mi>N</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>,</mo>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{2}\sim N(\mu _{2},\sigma _{2}^{2})}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2c1dacbf5aaa0ff0b9ade2992477d22baa23dcd" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.824ex; height:3.176ex;" alt="{\displaystyle X_{2}\sim N(\mu _{2},\sigma _{2}^{2})}"></span> is given by:<sup id="cite_ref-32" class="reference"><a href="#cite_note-32">&#91;32&#93;</a></sup> <div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{\mathrm {KL} }(X_{1}\parallel X_{2})={\frac {(\mu _{1}-\mu _{2})^{2}}{2\sigma _{2}^{2}}}+{\frac {1}{2}}\left({\frac {\sigma _{1}^{2}}{\sigma _{2}^{2}}}-1-\ln {\frac {\sigma _{1}^{2}}{\sigma _{2}^{2}}}\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>D</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">K</mi>
<mi mathvariant="normal">L</mi>
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</mrow>
</msub>
<mo stretchy="false">(</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>&#x2225;<!-- ∥ --></mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<msub>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
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</msub>
<mo>&#x2212;<!-- --></mo>
<msub>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msup>
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<mrow>
<mn>2</mn>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msubsup>
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</mfrac>
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<mo>+</mo>
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<mfrac>
<mn>1</mn>
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<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
</mfrac>
</mrow>
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
<mo>&#x2212;<!-- --></mo>
<mi>ln</mi>
<mo>&#x2061;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
</mfrac>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle D_{\mathrm {KL} }(X_{1}\parallel X_{2})={\frac {(\mu _{1}-\mu _{2})^{2}}{2\sigma _{2}^{2}}}+{\frac {1}{2}}\left({\frac {\sigma _{1}^{2}}{\sigma _{2}^{2}}}-1-\ln {\frac {\sigma _{1}^{2}}{\sigma _{2}^{2}}}\right)}</annotation>
</semantics>
</math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89a5fe7e76e6b9fbe30e34710d2d76d3073d89a6" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:53.751ex; height:7.509ex;" alt="{\displaystyle D_{\mathrm {KL} }(X_{1}\parallel X_{2})={\frac {(\mu _{1}-\mu _{2})^{2}}{2\sigma _{2}^{2}}}+{\frac {1}{2}}\left({\frac {\sigma _{1}^{2}}{\sigma _{2}^{2}}}-1-\ln {\frac {\sigma _{1}^{2}}{\sigma _{2}^{2}}}\right)}"></div>
The <a href="/wiki/Hellinger_distance" title="Hellinger distance">Hellinger distance</a> between the same distributions is equal to <div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{2}(X_{1},X_{2})=1-{\sqrt {\frac {2\sigma _{1}\sigma _{2}}{\sigma _{1}^{2}+\sigma _{2}^{2}}}}\exp \left(-{\frac {1}{4}}{\frac {(\mu _{1}-\mu _{2})^{2}}{\sigma _{1}^{2}+\sigma _{2}^{2}}}\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>H</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>1</mn>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mfrac>
<mrow>
<mn>2</mn>
<msub>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mrow>
<mrow>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
</mrow>
</mfrac>
</msqrt>
</mrow>
<mi>exp</mi>
<mo>&#x2061;<!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<msub>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>&#x2212;<!-- --></mo>
<msub>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mrow>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle H^{2}(X_{1},X_{2})=1-{\sqrt {\frac {2\sigma _{1}\sigma _{2}}{\sigma _{1}^{2}+\sigma _{2}^{2}}}}\exp \left(-{\frac {1}{4}}{\frac {(\mu _{1}-\mu _{2})^{2}}{\sigma _{1}^{2}+\sigma _{2}^{2}}}\right)}</annotation>
</semantics>
</math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b9f888884d0dcbcbb5ff916c32e1d2bad752517" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:52.708ex; height:7.843ex;" alt="{\displaystyle H^{2}(X_{1},X_{2})=1-{\sqrt {\frac {2\sigma _{1}\sigma _{2}}{\sigma _{1}^{2}+\sigma _{2}^{2}}}}\exp \left(-{\frac {1}{4}}{\frac {(\mu _{1}-\mu _{2})^{2}}{\sigma _{1}^{2}+\sigma _{2}^{2}}}\right)}"></div></li><li>The <a href="/wiki/Fisher_information_matrix" class="mw-redirect" title="Fisher information matrix">Fisher information matrix</a> for a normal distribution w.r.t. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="\mu "></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="\sigma ^{2}"></span> is diagonal and takes the form <div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {I}}(\mu ,\sigma ^{2})={\begin{pmatrix}{\frac {1}{\sigma ^{2}}}&amp;0\\0&amp;{\frac {1}{2\sigma ^{4}}}\end{pmatrix}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">I</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>,</mo>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>(</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mfrac>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {I}}(\mu ,\sigma ^{2})={\begin{pmatrix}{\frac {1}{\sigma ^{2}}}&amp;0\\0&amp;{\frac {1}{2\sigma ^{4}}}\end{pmatrix}}}</annotation>
</semantics>
</math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e882ba24b6d046a40e779c6154532c352ce59f35" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -3.505ex; margin-left: -0.069ex; width:24.469ex; height:8.176ex;" alt="{\displaystyle {\mathcal {I}}(\mu ,\sigma ^{2})={\begin{pmatrix}{\frac {1}{\sigma ^{2}}}&amp;0\\0&amp;{\frac {1}{2\sigma ^{4}}}\end{pmatrix}}}"></div></li><li>The <a href="/wiki/Conjugate_prior" title="Conjugate prior">conjugate prior</a> of the mean of a normal distribution is another normal distribution.<sup id="cite_ref-33" class="reference"><a href="#cite_note-33">&#91;33&#93;</a></sup> Specifically, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1},\ldots ,x_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>&#x2026;<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{1},\ldots ,x_{n}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/737e02a5fbf8bc31d443c91025339f9fd1de1065" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.11ex; height:2.009ex;" alt="x_{1},\ldots ,x_{n}"></span> are iid <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sim N(\mu ,\sigma ^{2})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>&#x223C;<!-- --></mo>
<mi>N</mi>
<mo stretchy="false">(</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>,</mo>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sim N(\mu ,\sigma ^{2})}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6360f79fbaa80195903d81ba400cc420cfe2045" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.147ex; height:3.176ex;" alt="{\displaystyle \sim N(\mu ,\sigma ^{2})}"></span> and the prior is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \sim N(\mu _{0},\sigma _{0}^{2})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
<mo>&#x223C;<!-- --></mo>
<mi>N</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>,</mo>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu \sim N(\mu _{0},\sigma _{0}^{2})}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c55a28f457a560eb30bb0d3bbdddf9eb3b6fc3b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.248ex; height:3.176ex;" alt="{\displaystyle \mu \sim N(\mu _{0},\sigma _{0}^{2})}"></span>, then the posterior distribution for the estimator of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="\mu "></span> will be <div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \mid x_{1},\ldots ,x_{n}\sim {\mathcal {N}}\left({\frac {{\frac {\sigma ^{2}}{n}}\mu _{0}+\sigma _{0}^{2}{\bar {x}}}{{\frac {\sigma ^{2}}{n}}+\sigma _{0}^{2}}},\left({\frac {n}{\sigma ^{2}}}+{\frac {1}{\sigma _{0}^{2}}}\right)^{-1}\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
<mo>&#x2223;<!-- --></mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>&#x2026;<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>&#x223C;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">N</mi>
</mrow>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>n</mi>
</mfrac>
</mrow>
<msub>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>+</mo>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">&#x00AF;<!-- ¯ --></mo>
</mover>
</mrow>
</mrow>
</mrow>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>n</mi>
</mfrac>
</mrow>
<mo>+</mo>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mrow>
<mo>,</mo>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>n</mi>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mfrac>
</mrow>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
</mfrac>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu \mid x_{1},\ldots ,x_{n}\sim {\mathcal {N}}\left({\frac {{\frac {\sigma ^{2}}{n}}\mu _{0}+\sigma _{0}^{2}{\bar {x}}}{{\frac {\sigma ^{2}}{n}}+\sigma _{0}^{2}}},\left({\frac {n}{\sigma ^{2}}}+{\frac {1}{\sigma _{0}^{2}}}\right)^{-1}\right)}</annotation>
</semantics>
</math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/793f7cd8d5c23e2f8a92fed1b332d89375f9229b" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:52.062ex; height:8.509ex;" alt="{\displaystyle \mu \mid x_{1},\ldots ,x_{n}\sim {\mathcal {N}}\left({\frac {{\frac {\sigma ^{2}}{n}}\mu _{0}+\sigma _{0}^{2}{\bar {x}}}{{\frac {\sigma ^{2}}{n}}+\sigma _{0}^{2}}},\left({\frac {n}{\sigma ^{2}}}+{\frac {1}{\sigma _{0}^{2}}}\right)^{-1}\right)}"></div></li><li>The family of normal distributions not only forms an <a href="/wiki/Exponential_family" title="Exponential family">exponential family</a> (EF), but in fact forms a <a href="/wiki/Natural_exponential_family" title="Natural exponential family">natural exponential family</a> (NEF) with quadratic <a href="/wiki/Variance_function" title="Variance function">variance function</a> (<a href="/wiki/NEF-QVF" class="mw-redirect" title="NEF-QVF">NEF-QVF</a>). Many properties of normal distributions generalize to properties of NEF-QVF distributions, NEF distributions, or EF distributions generally. NEF-QVF distributions comprises 6 families, including Poisson, Gamma, binomial, and negative binomial distributions, while many of the common families studied in probability and statistics are NEF or EF.</li><li>In <a href="/wiki/Information_geometry" title="Information geometry">information geometry</a>, the family of normal distributions forms a <a href="/wiki/Statistical_manifold" title="Statistical manifold">statistical manifold</a> with <a href="/wiki/Constant_curvature" title="Constant curvature">constant curvature</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle -1}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="-1"></span>. The same family is <a href="/wiki/Flat_manifold" title="Flat manifold">flat</a> with respect to the (±1)-connections <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla ^{(e)}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi mathvariant="normal">&#x2207;<!-- ∇ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mi>e</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \nabla ^{(e)}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9569f43424ae690f94543788b51f5cd729eb28d7" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.214ex; height:2.843ex;" alt="{\displaystyle \nabla ^{(e)}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla ^{(m)}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi mathvariant="normal">&#x2207;<!-- ∇ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mi>m</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \nabla ^{(m)}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19d030ffa365bc88e2a96130dda638bdc4877177" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.89ex; height:2.843ex;" alt="{\displaystyle \nabla ^{(m)}}"></span>.<sup id="cite_ref-34" class="reference"><a href="#cite_note-34">&#91;34&#93;</a></sup></li></ol></div>
<h2><span class="mw-headline" id="Related_distributions">Related distributions</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=21" title="Edit section: Related distributions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h2>
<h3><span class="mw-headline" id="Central_limit_theorem">Central limit theorem</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=22" title="Edit section: Central limit theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
<figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:De_moivre-laplace.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/0/06/De_moivre-laplace.gif" decoding="async" width="250" height="155" class="mw-file-element" data-file-width="250" data-file-height="155" /></a><figcaption>As the number of discrete events increases, the function begins to resemble a normal distribution</figcaption></figure>
<figure typeof="mw:File/Thumb"><a href="/wiki/File:Dice_sum_central_limit_theorem.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Dice_sum_central_limit_theorem.svg/250px-Dice_sum_central_limit_theorem.svg.png" decoding="async" width="250" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Dice_sum_central_limit_theorem.svg/375px-Dice_sum_central_limit_theorem.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Dice_sum_central_limit_theorem.svg/500px-Dice_sum_central_limit_theorem.svg.png 2x" data-file-width="512" data-file-height="614" /></a><figcaption>Comparison of probability density functions, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(k)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p(k)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f33a51ee10b4c7c54abdc5dbe61e358c7109308c" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.279ex; height:2.843ex;" alt="p(k)"></span> for the sum of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="n"></span> fair 6-sided dice to show their convergence to a normal distribution with increasing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle na}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mi>a</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle na}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92ad9df11220f7997992543b0053bcd211aadd78" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.625ex; height:1.676ex;" alt="{\displaystyle na}"></span>, in accordance to the central limit theorem. In the bottom-right graph, smoothed profiles of the previous graphs are rescaled, superimposed and compared with a normal distribution (black curve).</figcaption></figure>
<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1033289096"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Central_limit_theorem" title="Central limit theorem">Central limit theorem</a></div>
<p>The central limit theorem states that under certain (fairly common) conditions, the sum of many random variables will have an approximately normal distribution. More specifically, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},\ldots ,X_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>&#x2026;<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{1},\ldots ,X_{n}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac794f5521dcce89913085a6d566e7cdb615dbb0" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.299ex; height:2.509ex;" alt="{\displaystyle X_{1},\ldots ,X_{n}}"></span> are <a href="/wiki/Independent_and_identically_distributed" class="mw-redirect" title="Independent and identically distributed">independent and identically distributed</a> random variables with the same arbitrary distribution, zero mean, and variance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="\sigma ^{2}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Z</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="Z"></span> is their
mean scaled by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {n}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>n</mi>
</msqrt>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\sqrt {n}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a2994734eae382ce30100fb17b9447fd8e99f81" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.331ex; height:3.009ex;" alt="{\sqrt {n}}"></span>
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z={\sqrt {n}}\left({\frac {1}{n}}\sum _{i=1}^{n}X_{i}\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Z</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>n</mi>
</msqrt>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mi>n</mi>
</mfrac>
</mrow>
<munderover>
<mo>&#x2211;<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munderover>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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<mo>)</mo>
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</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Z={\sqrt {n}}\left({\frac {1}{n}}\sum _{i=1}^{n}X_{i}\right)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829b9ed598d6709ac090714bffab0cc7625507a8" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:21.262ex; height:7.509ex;" alt="Z={\sqrt {n}}\left({\frac {1}{n}}\sum _{i=1}^{n}X_{i}\right)"></span></dd></dl>
<p>Then, as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="n"></span> increases, the probability distribution of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Z</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="Z"></span> will tend to the normal distribution with zero mean and variance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="\sigma ^{2}"></span>.
</p><p>The theorem can be extended to variables <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X_{i})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (X_{i})}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67f488841364dd170c2f46faae8e2c3010f4cdb1" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.533ex; height:2.843ex;" alt="(X_{i})"></span> that are not independent and/or not identically distributed if certain constraints are placed on the degree of dependence and the moments of the distributions.
</p><p>Many <a href="/wiki/Test_statistic" title="Test statistic">test statistics</a>, <a href="/wiki/Score_(statistics)" class="mw-redirect" title="Score (statistics)">scores</a>, and <a href="/wiki/Estimator" title="Estimator">estimators</a> encountered in practice contain sums of certain random variables in them, and even more estimators can be represented as sums of random variables through the use of <a href="/wiki/Influence_function_(statistics)" class="mw-redirect" title="Influence function (statistics)">influence functions</a>. The central limit theorem implies that those statistical parameters will have asymptotically normal distributions.
</p><p>The central limit theorem also implies that certain distributions can be approximated by the normal distribution, for example:
</p>
<ul><li>The <a href="/wiki/Binomial_distribution" title="Binomial distribution">binomial distribution</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B(n,p)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>B</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>,</mo>
<mi>p</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle B(n,p)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58280f6b0f1a1b474a7047c07943f908e775aa71" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.171ex; height:2.843ex;" alt="B(n,p)"></span> is <a href="/wiki/De_Moivre%E2%80%93Laplace_theorem" title="De MoivreLaplace theorem">approximately normal</a> with mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle np}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mi>p</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle np}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d6eb41e0e5e136f594b1a703d2f371d9a5e0c27" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.564ex; height:2.009ex;" alt="np"></span> and variance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle np(1-p)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mi>p</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>&#x2212;<!-- --></mo>
<mi>p</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle np(1-p)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57f093250a1d822df677a03ac8aa78c6a8029866" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.546ex; height:2.843ex;" alt="np(1-p)"></span> for large <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="n"></span> and for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="p"></span> not too close to 0 or 1.</li>
<li>The <a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson distribution</a> with parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BB;<!-- λ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="\lambda "></span> is approximately normal with mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BB;<!-- λ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="\lambda "></span> and variance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BB;<!-- λ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="\lambda "></span>, for large values of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BB;<!-- λ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="\lambda "></span>.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35">&#91;35&#93;</a></sup></li>
<li>The <a href="/wiki/Chi-squared_distribution" title="Chi-squared distribution">chi-squared distribution</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi ^{2}(k)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>&#x03C7;<!-- χ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \chi ^{2}(k)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20d3f0ac864146e53b984408f3c9f75603d0e601" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.53ex; height:3.176ex;" alt="{\displaystyle \chi ^{2}(k)}"></span> is approximately normal with mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="k"></span> and variance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>2</mn>
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 2k}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab358eb7defb4d2b0fc1f9e8a4e2d189fe600eb6" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.374ex; height:2.176ex;" alt="2k"></span>, for large <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="k"></span>.</li>
<li>The <a href="/wiki/Student%27s_t-distribution" title="Student&#39;s t-distribution">Student's t-distribution</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t(\nu )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>t</mi>
<mo stretchy="false">(</mo>
<mi>&#x03BD;<!-- ν --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle t(\nu )}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/567c82d00fd3b45b8d5a8132780a186c62e605cb" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.881ex; height:2.843ex;" alt="{\displaystyle t(\nu )}"></span> is approximately normal with mean 0 and variance 1 when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BD;<!-- ν --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \nu }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="\nu "></span> is large.</li></ul>
<p>Whether these approximations are sufficiently accurate depends on the purpose for which they are needed, and the rate of convergence to the normal distribution. It is typically the case that such approximations are less accurate in the tails of the distribution.
</p><p>A general upper bound for the approximation error in the central limit theorem is given by the <a href="/wiki/Berry%E2%80%93Esseen_theorem" title="BerryEsseen theorem">BerryEsseen theorem</a>, improvements of the approximation are given by the <a href="/wiki/Edgeworth_expansion" class="mw-redirect" title="Edgeworth expansion">Edgeworth expansions</a>.
</p><p>This theorem can also be used to justify modeling the sum of many uniform noise sources as <a href="/wiki/Gaussian_noise" title="Gaussian noise">Gaussian noise</a>. See <a href="/wiki/AWGN" class="mw-redirect" title="AWGN">AWGN</a>.
</p>
<h3><span class="mw-headline" id="Operations_and_functions_of_normal_variables">Operations and functions of normal variables</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=23" title="Edit section: Operations and functions of normal variables"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
<figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Probabilities_of_functions_of_normal_vectors.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Probabilities_of_functions_of_normal_vectors.png/220px-Probabilities_of_functions_of_normal_vectors.png" decoding="async" width="220" height="797" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/Probabilities_of_functions_of_normal_vectors.png/330px-Probabilities_of_functions_of_normal_vectors.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/55/Probabilities_of_functions_of_normal_vectors.png/440px-Probabilities_of_functions_of_normal_vectors.png 2x" data-file-width="828" data-file-height="3001" /></a><figcaption><b>a:</b> Probability density of a function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos x^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>cos</mi>
<mo>&#x2061;<!-- --></mo>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \cos x^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc4e83e19d1c2de49bf5ce2d31395f1e0a04815d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.882ex; height:2.676ex;" alt="{\displaystyle \cos x^{2}}"></span> of a normal variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="x"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =-2}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
<mo>=</mo>
<mo>&#x2212;<!-- --></mo>
<mn>2</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu =-2}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8692ed3f07108ef682277ee64f17a81d49c74123" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.471ex; height:2.676ex;" alt="{\displaystyle \mu =-2}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma =3}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03C3;<!-- σ --></mi>
<mo>=</mo>
<mn>3</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma =3}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e868ad89842a9260535a342bca7cda8592b7e77" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle \sigma =3}"></span>. <b>b:</b> Probability density of a function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{y}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>y</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x^{y}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8561c712e86598255e8434a70affa18ffd7e0dd" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.379ex; height:2.343ex;" alt="x^y"></span> of two normal variables <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="x"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="y"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{x}=1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
</msub>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu _{x}=1}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27a203a4f0cfdb7a5d5a760a2ae186ac93d845c5" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.835ex; height:2.676ex;" alt="{\displaystyle \mu _{x}=1}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{y}=2}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>y</mi>
</mrow>
</msub>
<mo>=</mo>
<mn>2</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu _{y}=2}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b7bb907982ea07b174013055769e6a76f544441" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.712ex; height:2.843ex;" alt="{\displaystyle \mu _{y}=2}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{x}=0.1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
</msub>
<mo>=</mo>
<mn>0.1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma _{x}=0.1}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5da3e9fc63055fac514f83a129f223396e6ec60" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.57ex; height:2.509ex;" alt="{\displaystyle \sigma _{x}=0.1}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{y}=0.2}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>y</mi>
</mrow>
</msub>
<mo>=</mo>
<mn>0.2</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma _{y}=0.2}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98c9d8696dfec56572f0e4ac5ef664a50ff63114" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.447ex; height:2.843ex;" alt="{\displaystyle \sigma _{y}=0.2}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{xy}=0.8}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>&#x03C1;<!-- ρ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
<mi>y</mi>
</mrow>
</msub>
<mo>=</mo>
<mn>0.8</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \rho _{xy}=0.8}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04fb480f8ce9e65dda6f87ad8d454b50b679b75d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.262ex; height:2.843ex;" alt="{\displaystyle \rho _{xy}=0.8}"></span>. <b>c:</b> Heat map of the joint probability density of two functions of two correlated normal variables <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="x"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle y}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="y"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{x}=-2}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
</msub>
<mo>=</mo>
<mo>&#x2212;<!-- --></mo>
<mn>2</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu _{x}=-2}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e29a8e49c5b014588384678588a61cc2cea5cba0" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.643ex; height:2.676ex;" alt="{\displaystyle \mu _{x}=-2}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{y}=5}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>y</mi>
</mrow>
</msub>
<mo>=</mo>
<mn>5</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu _{y}=5}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fdb724b9cf16699abd76b3d785654df84e7c319" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.712ex; height:2.843ex;" alt="{\displaystyle \mu _{y}=5}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{x}^{2}=10}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>=</mo>
<mn>10</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma _{x}^{2}=10}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/821007ad6d915b39d6b75bd2efe87269f41ecd50" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.923ex; height:2.843ex;" alt="{\displaystyle \sigma _{x}^{2}=10}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{y}^{2}=20}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>y</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>=</mo>
<mn>20</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma _{y}^{2}=20}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4285f9fb6dc81c6f3f7a0ecb0ca83c0d59133810" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.808ex; height:3.176ex;" alt="{\displaystyle \sigma _{y}^{2}=20}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{xy}=0.495}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>&#x03C1;<!-- ρ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>x</mi>
<mi>y</mi>
</mrow>
</msub>
<mo>=</mo>
<mn>0.495</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \rho _{xy}=0.495}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8aa55d8c9a06d50a54d9b9c51ca1a4ebaabf339c" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.587ex; height:2.843ex;" alt="{\displaystyle \rho _{xy}=0.495}"></span>. <b>d:</b> Probability density of a function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{i=1}^{4}\vert x_{i}\vert }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<munderover>
<mo>&#x2211;<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</munderover>
<mo fence="false" stretchy="false">|</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo fence="false" stretchy="false">|</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\textstyle \sum _{i=1}^{4}\vert x_{i}\vert }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04629b53bb6deeededd41584e7e320bc6deb24f0" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.164ex; height:3.509ex;" alt="{\textstyle \sum _{i=1}^{4}\vert x_{i}\vert }"></span> of 4 iid standard normal variables. These are computed by the numerical method of ray-tracing.<sup id="cite_ref-Das_36-0" class="reference"><a href="#cite_note-Das-36">&#91;36&#93;</a></sup></figcaption></figure>
<p>The <a href="/wiki/Probability_density_function" title="Probability density function">probability density</a>, <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution</a>, and <a href="/wiki/Inverse_cumulative_distribution_function" class="mw-redirect" title="Inverse cumulative distribution function">inverse cumulative distribution</a> of any function of one or more independent or correlated normal variables can be computed with the numerical method of ray-tracing<sup id="cite_ref-Das_36-1" class="reference"><a href="#cite_note-Das-36">&#91;36&#93;</a></sup> (<a rel="nofollow" class="external text" href="https://www.mathworks.com/matlabcentral/fileexchange/84973-integrate-and-classify-normal-distributions">Matlab code</a>). In the following sections we look at some special cases.
</p>
<h4><span class="mw-headline" id="Operations_on_a_single_normal_variable">Operations on a single normal variable</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=24" title="Edit section: Operations on a single normal variable"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h4>
<p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"></span> is distributed normally with mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="\mu "></span> and variance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="\sigma ^{2}"></span>, then
</p>
<ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle aX+b}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
<mi>X</mi>
<mo>+</mo>
<mi>b</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle aX+b}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/216cce462e394ea3411f94820bcb8cc4660a1bb0" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.048ex; height:2.343ex;" alt="{\displaystyle aX+b}"></span>, for any real numbers <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="a"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>b</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="b"></span>, is also normally distributed, with mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\mu +b}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
<mi>&#x03BC;<!-- μ --></mi>
<mo>+</mo>
<mi>b</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a\mu +b}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3cc43983d0a0fdf489677b407f11fa06cb47c5b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.469ex; height:2.676ex;" alt="{\displaystyle a\mu +b}"></span> and standard deviation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |a|\sigma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>&#x03C3;<!-- σ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |a|\sigma }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e542d90598717553013ab5a4ae7e9c5e7a53f7c" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.853ex; height:2.843ex;" alt="{\displaystyle |a|\sigma }"></span>. That is, the family of normal distributions is closed under <a href="/wiki/Linear_map" title="Linear map">linear transformations</a>.</li>
<li>The exponential of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"></span> is distributed <a href="/wiki/Log-normal_distribution" title="Log-normal distribution">log-normally</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{X}\sim \ln(N(\mu ,\sigma ^{2}))}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>X</mi>
</mrow>
</msup>
<mo>&#x223C;<!-- --></mo>
<mi>ln</mi>
<mo>&#x2061;<!-- --></mo>
<mo stretchy="false">(</mo>
<mi>N</mi>
<mo stretchy="false">(</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>,</mo>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle e^{X}\sim \ln(N(\mu ,\sigma ^{2}))}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5037cfa5aef3735e9af2e1748f0e18b8494c27da" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.256ex; height:3.176ex;" alt="{\displaystyle e^{X}\sim \ln(N(\mu ,\sigma ^{2}))}"></span>.</li>
<li>The absolute value of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"></span> has <a href="/wiki/Folded_normal_distribution" title="Folded normal distribution">folded normal distribution</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\left|X\right|\sim N_{f}(\mu ,\sigma ^{2})}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>|</mo>
<mi>X</mi>
<mo>|</mo>
</mrow>
<mo>&#x223C;<!-- --></mo>
<msub>
<mi>N</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>f</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>,</mo>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\left|X\right|\sim N_{f}(\mu ,\sigma ^{2})}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/477baf07689f2d7caa80b058dc456055964cc64f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.004ex; height:3.343ex;" alt="{\displaystyle {\left|X\right|\sim N_{f}(\mu ,\sigma ^{2})}}"></span>. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu =0}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3753282c0ad2ea1e7d63f39425efd13c37da3169" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.663ex; height:2.676ex;" alt="\mu =0"></span> this is known as the <a href="/wiki/Half-normal_distribution" title="Half-normal distribution">half-normal distribution</a>.</li>
<li>The absolute value of normalized residuals, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |X-\mu |/\sigma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>X</mi>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>&#x03C3;<!-- σ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |X-\mu |/\sigma }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e9ed6377872342ec188ccb2a6b09889251d1e12" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.008ex; height:2.843ex;" alt="{\displaystyle |X-\mu |/\sigma }"></span>, has <a href="/wiki/Chi_distribution" title="Chi distribution">chi distribution</a> with one degree of freedom: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |X-\mu |/\sigma \sim \chi _{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>X</mi>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>&#x03C3;<!-- σ --></mi>
<mo>&#x223C;<!-- --></mo>
<msub>
<mi>&#x03C7;<!-- χ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |X-\mu |/\sigma \sim \chi _{1}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/891165a7f7378e5cd630e5b53d82fa2a465aa450" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.616ex; height:2.843ex;" alt="{\displaystyle |X-\mu |/\sigma \sim \chi _{1}}"></span>.</li>
<li>The square of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X/\sigma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>&#x03C3;<!-- σ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X/\sigma }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a232c0ac9bf7fd54eab97a4e11bdf6ca55edd728" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.472ex; height:2.843ex;" alt="{\displaystyle X/\sigma }"></span> has the <a href="/wiki/Noncentral_chi-squared_distribution" title="Noncentral chi-squared distribution">noncentral chi-squared distribution</a> with one degree of freedom: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle X^{2}/\sigma ^{2}\sim \chi _{1}^{2}(\mu ^{2}/\sigma ^{2})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<msup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>&#x223C;<!-- --></mo>
<msubsup>
<mi>&#x03C7;<!-- χ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<msup>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\textstyle X^{2}/\sigma ^{2}\sim \chi _{1}^{2}(\mu ^{2}/\sigma ^{2})}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8b3d087928e23e018e2e2e7ff3badaf23bdb536" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.018ex; height:3.176ex;" alt="{\textstyle X^{2}/\sigma ^{2}\sim \chi _{1}^{2}(\mu ^{2}/\sigma ^{2})}"></span>. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu =0}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3753282c0ad2ea1e7d63f39425efd13c37da3169" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.663ex; height:2.676ex;" alt="\mu =0"></span>, the distribution is called simply <a href="/wiki/Chi-squared_distribution" title="Chi-squared distribution">chi-squared</a>.</li>
<li>The log-likelihood of a normal variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="x"></span> is simply the log of its <a href="/wiki/Probability_density_function" title="Probability density function">probability density function</a>: <div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln p(x)=-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}-\ln \left(\sigma {\sqrt {2\pi }}\right).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ln</mi>
<mo>&#x2061;<!-- --></mo>
<mi>p</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>x</mi>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
</mrow>
<mi>&#x03C3;<!-- σ --></mi>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>&#x2212;<!-- --></mo>
<mi>ln</mi>
<mo>&#x2061;<!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>2</mn>
<mi>&#x03C0;<!-- π --></mi>
</msqrt>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \ln p(x)=-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}-\ln \left(\sigma {\sqrt {2\pi }}\right).}</annotation>
</semantics>
</math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de1b937be2248646d42f58a296f05ec46eab9ded" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.74ex; height:6.509ex;" alt="{\displaystyle \ln p(x)=-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}-\ln \left(\sigma {\sqrt {2\pi }}\right).}"></div> Since this is a scaled and shifted square of a standard normal variable, it is distributed as a scaled and shifted <a href="/wiki/Chi-squared_distribution" title="Chi-squared distribution">chi-squared</a> variable.</li>
<li>The distribution of the variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"></span> restricted to an interval <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">[</mo>
<mi>a</mi>
<mo>,</mo>
<mi>b</mi>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="[a,b]"></span> is called the <a href="/wiki/Truncated_normal_distribution" title="Truncated normal distribution">truncated normal distribution</a>.</li>
<li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X-\mu )^{-2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>X</mi>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (X-\mu )^{-2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2371268d360d0d951fb201cf495d3a86c983a319" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.364ex; height:3.176ex;" alt="{\displaystyle (X-\mu )^{-2}}"></span> has a <a href="/wiki/L%C3%A9vy_distribution" title="Lévy distribution">Lévy distribution</a> with location 0 and scale <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{-2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma ^{-2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/344c690cb6e4e0c32cf8c971a8e4823a4ed935fe" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.663ex; height:2.676ex;" alt="{\displaystyle \sigma ^{-2}}"></span>.</li></ul>
<h5><span class="mw-headline" id="Operations_on_two_independent_normal_variables">Operations on two independent normal variables</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=25" title="Edit section: Operations on two independent normal variables"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h5>
<ul><li>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{1}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f70b2694445a5901b24338a2e7a7e58f02a72a32" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="X_{1}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ad47c14b8a092f182512e76c96638aea6e3bea1" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="X_{2}"></span> are two <a href="/wiki/Independence_(probability_theory)" title="Independence (probability theory)">independent</a> normal random variables, with means <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu _{1}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6899621035d3359b9c1c064739b54c7004e220d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.456ex; height:2.176ex;" alt="\mu _{1}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu _{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f841461ae8f2eafec3fe879f7c061a73c2f7170" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.456ex; height:2.176ex;" alt="\mu _{2}"></span> and standard deviations <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma _{1}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fa0e56273a1cb32709b442e2421e9f947522b84" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.382ex; height:2.009ex;" alt="\sigma _{1}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma _{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d4b9cd9efc54bcfd04e0a2231913c13f10798d9" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.382ex; height:2.009ex;" alt="\sigma _{2}"></span>, then their sum <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}+X_{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{1}+X_{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f96d604ba472d9c8d3964bfb198de16b68e5f8a3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.797ex; height:2.509ex;" alt="X_{1}+X_{2}"></span> will also be normally distributed,<sup><a href="/wiki/Sum_of_normally_distributed_random_variables" title="Sum of normally distributed random variables">[proof]</a></sup> with mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{1}+\mu _{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu _{1}+\mu _{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c63b3f9c1c00ce3688f617bbecaeb5137de58f58" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.752ex; height:2.509ex;" alt="{\displaystyle \mu _{1}+\mu _{2}}"></span> and variance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{1}^{2}+\sigma _{2}^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma _{1}^{2}+\sigma _{2}^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0ce72b6024d3eabb8fda0dc65c8efa8dbe90bb6" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.61ex; height:3.176ex;" alt="\sigma _{1}^{2}+\sigma _{2}^{2}"></span>.</li>
<li>In particular, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="Y"></span> are independent normal deviates with zero mean and variance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="\sigma ^{2}"></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X+Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>+</mo>
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X+Y}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/191744cf9cddeff3ab2e750e22bcfce7766d355e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.594ex; height:2.343ex;" alt="X+Y"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X-Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>&#x2212;<!-- --></mo>
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X-Y}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08754e521ae59759cd7ed7dc8b5c73c8b931f16a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.594ex; height:2.343ex;" alt="X-Y"></span> are also independent and normally distributed, with zero mean and variance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\sigma ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>2</mn>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 2\sigma ^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3232261065274c69d6f2d81dd6aaf06f44922aa" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.547ex; height:2.676ex;" alt="2\sigma ^{2}"></span>. This is a special case of the <a href="/wiki/Polarization_identity" title="Polarization identity">polarization identity</a>.<sup id="cite_ref-37" class="reference"><a href="#cite_note-37">&#91;37&#93;</a></sup></li>
<li>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{1}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f70b2694445a5901b24338a2e7a7e58f02a72a32" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="X_{1}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ad47c14b8a092f182512e76c96638aea6e3bea1" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="X_{2}"></span> are two independent normal deviates with mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="\mu "></span> and deviation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03C3;<!-- σ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="\sigma "></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="a"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>b</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle b}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="b"></span> are arbitrary real numbers, then the variable <div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{3}={\frac {aX_{1}+bX_{2}-(a+b)\mu }{\sqrt {a^{2}+b^{2}}}}+\mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>a</mi>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<mi>b</mi>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>&#x2212;<!-- --></mo>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
<mo stretchy="false">)</mo>
<mi>&#x03BC;<!-- μ --></mi>
</mrow>
<msqrt>
<msup>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>+</mo>
<msup>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</msqrt>
</mfrac>
</mrow>
<mo>+</mo>
<mi>&#x03BC;<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{3}={\frac {aX_{1}+bX_{2}-(a+b)\mu }{\sqrt {a^{2}+b^{2}}}}+\mu }</annotation>
</semantics>
</math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cecad53efc9fb1f034f57b6ca0dd5754f504c919" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:33.299ex; height:7.009ex;" alt="{\displaystyle X_{3}={\frac {aX_{1}+bX_{2}-(a+b)\mu }{\sqrt {a^{2}+b^{2}}}}+\mu }"></div> is also normally distributed with mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="\mu "></span> and deviation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03C3;<!-- σ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="\sigma "></span>. It follows that the normal distribution is <a href="/wiki/Stable_distribution" title="Stable distribution">stable</a> (with exponent <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =2}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03B1;<!-- α --></mi>
<mo>=</mo>
<mn>2</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \alpha =2}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/938489e6428bb7959330df8c06c79a994811c4a9" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="\alpha =2"></span>).</li>
<li>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{k}\sim {\mathcal {N}}(m_{k},\sigma _{k}^{2})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo>&#x223C;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">N</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<msub>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo>,</mo>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{k}\sim {\mathcal {N}}(m_{k},\sigma _{k}^{2})}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa00416767bc695f711a6a1c64952d9590292407" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.775ex; height:3.176ex;" alt="{\displaystyle X_{k}\sim {\mathcal {N}}(m_{k},\sigma _{k}^{2})}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \{0,1\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
<mo>&#x2208;<!-- ∈ --></mo>
<mo fence="false" stretchy="false">{</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k\in \{0,1\}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b9ce97a81a623b271691ba41ecefd48c9a7e69" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.736ex; height:2.843ex;" alt="{\displaystyle k\in \{0,1\}}"></span> are normal distributions, then their normalized <a href="/wiki/Geometric_mean" title="Geometric mean">geometric mean</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\int _{\mathbb {R} ^{n}}X_{0}^{\alpha }(x)X_{1}^{1-\alpha }(x)\,{\text{d}}x}}X_{0}^{\alpha }X_{1}^{1-\alpha }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<msub>
<mo>&#x222B;<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mrow>
</msub>
<msubsup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>&#x03B1;<!-- α --></mi>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<msubsup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03B1;<!-- α --></mi>
</mrow>
</msubsup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace" />
<mrow class="MJX-TeXAtom-ORD">
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<msubsup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>&#x03B1;<!-- α --></mi>
</mrow>
</msubsup>
<msubsup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03B1;<!-- α --></mi>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\int _{\mathbb {R} ^{n}}X_{0}^{\alpha }(x)X_{1}^{1-\alpha }(x)\,{\text{d}}x}}X_{0}^{\alpha }X_{1}^{1-\alpha }}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/877409a8811b0b9624918b7a9f0a717e92d42c48" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.317ex; height:6.509ex;" alt="{\displaystyle {\frac {1}{\int _{\mathbb {R} ^{n}}X_{0}^{\alpha }(x)X_{1}^{1-\alpha }(x)\,{\text{d}}x}}X_{0}^{\alpha }X_{1}^{1-\alpha }}"></span> is a normal distribution <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}(m_{\alpha },\sigma _{\alpha }^{2})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">N</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<msub>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>&#x03B1;<!-- α --></mi>
</mrow>
</msub>
<mo>,</mo>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>&#x03B1;<!-- α --></mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}(m_{\alpha },\sigma _{\alpha }^{2})}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a40c2d224aed4998bdf5faee0fd98f04b701e94f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.062ex; width:11.116ex; height:3.009ex;" alt="{\displaystyle {\mathcal {N}}(m_{\alpha },\sigma _{\alpha }^{2})}"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{\alpha }={\frac {\alpha m_{0}\sigma _{1}^{2}+(1-\alpha )m_{1}\sigma _{0}^{2}}{\alpha \sigma _{1}^{2}+(1-\alpha )\sigma _{0}^{2}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>&#x03B1;<!-- α --></mi>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>&#x03B1;<!-- α --></mi>
<msub>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03B1;<!-- α --></mi>
<mo stretchy="false">)</mo>
<msub>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
</mrow>
<mrow>
<mi>&#x03B1;<!-- α --></mi>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03B1;<!-- α --></mi>
<mo stretchy="false">)</mo>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle m_{\alpha }={\frac {\alpha m_{0}\sigma _{1}^{2}+(1-\alpha )m_{1}\sigma _{0}^{2}}{\alpha \sigma _{1}^{2}+(1-\alpha )\sigma _{0}^{2}}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e33afc66e4f4e9355877edfac17bb66d5aec40d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:29.846ex; height:7.176ex;" alt="{\displaystyle m_{\alpha }={\frac {\alpha m_{0}\sigma _{1}^{2}+(1-\alpha )m_{1}\sigma _{0}^{2}}{\alpha \sigma _{1}^{2}+(1-\alpha )\sigma _{0}^{2}}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{\alpha }^{2}={\frac {\sigma _{0}^{2}\sigma _{1}^{2}}{\alpha \sigma _{1}^{2}+(1-\alpha )\sigma _{0}^{2}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>&#x03B1;<!-- α --></mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
</mrow>
<mrow>
<mi>&#x03B1;<!-- α --></mi>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03B1;<!-- α --></mi>
<mo stretchy="false">)</mo>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma _{\alpha }^{2}={\frac {\sigma _{0}^{2}\sigma _{1}^{2}}{\alpha \sigma _{1}^{2}+(1-\alpha )\sigma _{0}^{2}}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40ffb2767801c70420c49b60e1166914a1696706" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.943ex; height:7.176ex;" alt="{\displaystyle \sigma _{\alpha }^{2}={\frac {\sigma _{0}^{2}\sigma _{1}^{2}}{\alpha \sigma _{1}^{2}+(1-\alpha )\sigma _{0}^{2}}}}"></span> (see <a rel="nofollow" class="external text" href="https://www.geogebra.org/geometry/qfqnjtsu">here</a> for a visualization).</li></ul>
<h5><span class="mw-headline" id="Operations_on_two_independent_standard_normal_variables">Operations on two independent standard normal variables</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=26" title="Edit section: Operations on two independent standard normal variables"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h5>
<p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{1}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f70b2694445a5901b24338a2e7a7e58f02a72a32" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="X_{1}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ad47c14b8a092f182512e76c96638aea6e3bea1" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="X_{2}"></span> are two independent standard normal random variables with mean 0 and variance 1, then
</p>
<ul><li>Their sum and difference is distributed normally with mean zero and variance two: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}\pm X_{2}\sim {\mathcal {N}}(0,2)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>&#x00B1;<!-- ± --></mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>&#x223C;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">N</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>,</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{1}\pm X_{2}\sim {\mathcal {N}}(0,2)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c22f2550b85fe15bc71e1a7930b23d01e79d606a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.339ex; height:3.009ex;" alt="{\displaystyle X_{1}\pm X_{2}\sim {\mathcal {N}}(0,2)}"></span>.</li>
<li>Their product <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z=X_{1}X_{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Z</mi>
<mo>=</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Z=X_{1}X_{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58f0f568afbff4a886769be26450f6c6f560d3f2" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.736ex; height:2.509ex;" alt="{\displaystyle Z=X_{1}X_{2}}"></span> follows the <a href="/wiki/Product_distribution#Independent_central-normal_distributions" class="mw-redirect" title="Product distribution">product distribution</a><sup id="cite_ref-38" class="reference"><a href="#cite_note-38">&#91;38&#93;</a></sup> with density function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{Z}(z)=\pi ^{-1}K_{0}(|z|)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>Z</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mi>&#x03C0;<!-- π --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
</mrow>
</msup>
<msub>
<mi>K</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f_{Z}(z)=\pi ^{-1}K_{0}(|z|)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/077879f46bfe09ba1b2d43b61799a487efcc404e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.441ex; height:3.176ex;" alt="{\displaystyle f_{Z}(z)=\pi ^{-1}K_{0}(|z|)}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>K</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle K_{0}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44b0af6cafb690d3dbb0f3f30a032631338dc476" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.027ex; height:2.509ex;" alt="K_{0}"></span> is the <a href="/wiki/Macdonald_function" class="mw-redirect" title="Macdonald function">modified Bessel function of the second kind</a>. This distribution is symmetric around zero, unbounded at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>z</mi>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle z=0}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b92bfc06485cc90286474b14a516a68d8bfdd7b3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.349ex; height:2.176ex;" alt="z=0"></span>, and has the <a href="/wiki/Characteristic_function_(probability_theory)" title="Characteristic function (probability theory)">characteristic function</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{Z}(t)=(1+t^{2})^{-1/2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>&#x03D5;<!-- ϕ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>Z</mi>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \phi _{Z}(t)=(1+t^{2})^{-1/2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60630ad89912de003562901d097309fb387d5044" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.236ex; height:3.343ex;" alt="{\displaystyle \phi _{Z}(t)=(1+t^{2})^{-1/2}}"></span>.</li>
<li>Their ratio follows the standard <a href="/wiki/Cauchy_distribution" title="Cauchy distribution">Cauchy distribution</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}/X_{2}\sim \operatorname {Cauchy} (0,1)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>&#x223C;<!-- --></mo>
<mi>Cauchy</mi>
<mo>&#x2061;<!-- --></mo>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{1}/X_{2}\sim \operatorname {Cauchy} (0,1)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6535a528ac8f45d7a0c45f376cb44fce92fd2b43" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.072ex; height:2.843ex;" alt="{\displaystyle X_{1}/X_{2}\sim \operatorname {Cauchy} (0,1)}"></span>.</li>
<li>Their Euclidean norm <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {X_{1}^{2}+X_{2}^{2}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<msubsup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
</msqrt>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\sqrt {X_{1}^{2}+X_{2}^{2}}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53c70b7925645b1f55afdec8c20b49137c77084c" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.266ex; height:4.843ex;" alt="{\displaystyle {\sqrt {X_{1}^{2}+X_{2}^{2}}}}"></span> has the <a href="/wiki/Rayleigh_distribution" title="Rayleigh distribution">Rayleigh distribution</a>.</li></ul>
<h4><span class="mw-headline" id="Operations_on_multiple_independent_normal_variables">Operations on multiple independent normal variables</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=27" title="Edit section: Operations on multiple independent normal variables"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h4>
<ul><li>Any <a href="/wiki/Linear_combination" title="Linear combination">linear combination</a> of independent normal deviates is a normal deviate.</li>
<li>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},X_{2},\ldots ,X_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>&#x2026;<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2},\ldots ,X_{n}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d67872301909a9d739e265252ad0c7339cead069" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.312ex; height:2.509ex;" alt="X_{1},X_{2},\ldots ,X_{n}"></span> are independent standard normal random variables, then the sum of their squares has the <a href="/wiki/Chi-squared_distribution" title="Chi-squared distribution">chi-squared distribution</a> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="n"></span> degrees of freedom <div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}^{2}+\cdots +X_{n}^{2}\sim \chi _{n}^{2}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>+</mo>
<mo>&#x22EF;<!-- ⋯ --></mo>
<mo>+</mo>
<msubsup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>&#x223C;<!-- --></mo>
<msubsup>
<mi>&#x03C7;<!-- χ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{1}^{2}+\cdots +X_{n}^{2}\sim \chi _{n}^{2}.}</annotation>
</semantics>
</math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b76ecc4420156e1818fea4a8500786c0cc41ecd" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.017ex; height:3.176ex;" alt="{\displaystyle X_{1}^{2}+\cdots +X_{n}^{2}\sim \chi _{n}^{2}.}"></div></li>
<li>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},X_{2},\ldots ,X_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>&#x2026;<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2},\ldots ,X_{n}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d67872301909a9d739e265252ad0c7339cead069" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.312ex; height:2.509ex;" alt="X_{1},X_{2},\ldots ,X_{n}"></span> are independent normally distributed random variables with means <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="\mu "></span> and variances <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="\sigma ^{2}"></span>, then their <a href="/wiki/Sample_mean" class="mw-redirect" title="Sample mean">sample mean</a> is independent from the sample <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviation</a>,<sup id="cite_ref-39" class="reference"><a href="#cite_note-39">&#91;39&#93;</a></sup> which can be demonstrated using <a href="/wiki/Basu%27s_theorem" title="Basu&#39;s theorem">Basu's theorem</a> or <a href="/wiki/Cochran%27s_theorem" title="Cochran&#39;s theorem">Cochran's theorem</a>.<sup id="cite_ref-40" class="reference"><a href="#cite_note-40">&#91;40&#93;</a></sup> The ratio of these two quantities will have the <a href="/wiki/Student%27s_t-distribution" title="Student&#39;s t-distribution">Student's t-distribution</a> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n-1}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd0b0f32b28f51962943ee9ede4fb34198a2521" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="n-1"></span> degrees of freedom: <div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t={\frac {{\overline {X}}-\mu }{S/{\sqrt {n}}}}={\frac {{\frac {1}{n}}(X_{1}+\cdots +X_{n})-\mu }{\sqrt {{\frac {1}{n(n-1)}}\left[(X_{1}-{\overline {X}})^{2}+\cdots +(X_{n}-{\overline {X}})^{2}\right]}}}\sim t_{n-1}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>t</mi>
<mo>=</mo>
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<mo>&#x2212;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
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<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
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<mrow class="MJX-TeXAtom-ORD">
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<mo>+</mo>
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<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
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<mo stretchy="false">)</mo>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
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<msqrt>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mi>n</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
<mrow>
<mo>[</mo>
<mrow>
<mo stretchy="false">(</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
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</msub>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>X</mi>
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<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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<mo>+</mo>
<mo>&#x22EF;<!-- ⋯ --></mo>
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<mo stretchy="false">(</mo>
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<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
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<mo>&#x2212;<!-- --></mo>
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<mover>
<mi>X</mi>
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<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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<mo>]</mo>
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<mo>&#x223C;<!-- --></mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
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<mo>.</mo>
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<annotation encoding="application/x-tex">{\displaystyle t={\frac {{\overline {X}}-\mu }{S/{\sqrt {n}}}}={\frac {{\frac {1}{n}}(X_{1}+\cdots +X_{n})-\mu }{\sqrt {{\frac {1}{n(n-1)}}\left[(X_{1}-{\overline {X}})^{2}+\cdots +(X_{n}-{\overline {X}})^{2}\right]}}}\sim t_{n-1}.}</annotation>
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</math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36ff0d3c79a0504e8f259ef99192b825357914d7" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:64.276ex; height:10.343ex;" alt="{\displaystyle t={\frac {{\overline {X}}-\mu }{S/{\sqrt {n}}}}={\frac {{\frac {1}{n}}(X_{1}+\cdots +X_{n})-\mu }{\sqrt {{\frac {1}{n(n-1)}}\left[(X_{1}-{\overline {X}})^{2}+\cdots +(X_{n}-{\overline {X}})^{2}\right]}}}\sim t_{n-1}.}"></div></li>
<li>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},X_{2},\ldots ,X_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>&#x2026;<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{1},X_{2},\ldots ,X_{n}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d67872301909a9d739e265252ad0c7339cead069" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.312ex; height:2.509ex;" alt="X_{1},X_{2},\ldots ,X_{n}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y_{1},Y_{2},\ldots ,Y_{m}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>&#x2026;<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y_{1},Y_{2},\ldots ,Y_{m}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f02ae42b54857778de8e959388cd4d3d7641cce0" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.047ex; height:2.509ex;" alt="{\displaystyle Y_{1},Y_{2},\ldots ,Y_{m}}"></span> are independent standard normal random variables, then the ratio of their normalized sums of squares will have the <span class="nowrap"><a href="/wiki/F-distribution" title="F-distribution">F-distribution</a></span> with <span class="texhtml">(<i>n</i>, <i>m</i>)</span> degrees of freedom:<sup id="cite_ref-41" class="reference"><a href="#cite_note-41">&#91;41&#93;</a></sup> <div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F={\frac {\left(X_{1}^{2}+X_{2}^{2}+\cdots +X_{n}^{2}\right)/n}{\left(Y_{1}^{2}+Y_{2}^{2}+\cdots +Y_{m}^{2}\right)/m}}\sim F_{n,m}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<msubsup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
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<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msubsup>
<mo>+</mo>
<msubsup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>+</mo>
<mo>&#x22EF;<!-- ⋯ --></mo>
<mo>+</mo>
<msubsup>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msubsup>
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<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>n</mi>
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<mrow>
<mo>(</mo>
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<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
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<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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<mo>+</mo>
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<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msubsup>
<mo>+</mo>
<mo>&#x22EF;<!-- ⋯ --></mo>
<mo>+</mo>
<msubsup>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>m</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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<mo>)</mo>
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<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>m</mi>
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</mfrac>
</mrow>
<mo>&#x223C;<!-- --></mo>
<msub>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>,</mo>
<mi>m</mi>
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<mo>.</mo>
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<annotation encoding="application/x-tex">{\displaystyle F={\frac {\left(X_{1}^{2}+X_{2}^{2}+\cdots +X_{n}^{2}\right)/n}{\left(Y_{1}^{2}+Y_{2}^{2}+\cdots +Y_{m}^{2}\right)/m}}\sim F_{n,m}.}</annotation>
</semantics>
</math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c1b5d2ab40c3e85b5f24d5b13e8f95202fdca93" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:39.933ex; height:7.343ex;" alt="{\displaystyle F={\frac {\left(X_{1}^{2}+X_{2}^{2}+\cdots +X_{n}^{2}\right)/n}{\left(Y_{1}^{2}+Y_{2}^{2}+\cdots +Y_{m}^{2}\right)/m}}\sim F_{n,m}.}"></div></li></ul>
<h4><span class="mw-headline" id="Operations_on_multiple_correlated_normal_variables">Operations on multiple correlated normal variables</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=28" title="Edit section: Operations on multiple correlated normal variables"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h4>
<ul><li>A <a href="/wiki/Quadratic_form" title="Quadratic form">quadratic form</a> of a normal vector, i.e. a quadratic function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle q=\sum x_{i}^{2}+\sum x_{j}+c}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mi>q</mi>
<mo>=</mo>
<mo>&#x2211;<!-- ∑ --></mo>
<msubsup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>+</mo>
<mo>&#x2211;<!-- ∑ --></mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo>+</mo>
<mi>c</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\textstyle q=\sum x_{i}^{2}+\sum x_{j}+c}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ed0ec175c34bf70d7336eb9f25dbc2d269ff701" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.161ex; height:3.176ex;" alt="{\textstyle q=\sum x_{i}^{2}+\sum x_{j}+c}"></span> of multiple independent or correlated normal variables, is a <a href="/wiki/Generalized_chi-square_distribution" class="mw-redirect" title="Generalized chi-square distribution">generalized chi-square</a> variable.</li></ul>
<h3><span class="mw-headline" id="Operations_on_the_density_function">Operations on the density function</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=29" title="Edit section: Operations on the density function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
<p>The <a href="/wiki/Split_normal_distribution" title="Split normal distribution">split normal distribution</a> is most directly defined in terms of joining scaled sections of the density functions of different normal distributions and rescaling the density to integrate to one. The <a href="/wiki/Truncated_normal_distribution" title="Truncated normal distribution">truncated normal distribution</a> results from rescaling a section of a single density function.
</p>
<h3><span id="Infinite_divisibility_and_Cram.C3.A9r.27s_theorem"></span><span class="mw-headline" id="Infinite_divisibility_and_Cramér's_theorem">Infinite divisibility and Cramér's theorem</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=30" title="Edit section: Infinite divisibility and Cramér&#039;s theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
<p>For any positive integer <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{n}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtext>n</mtext>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\text{n}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eaaf79c97c17e3d4d0a55bc13e965bacfbff279e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:1.676ex;" alt="{\displaystyle {\text{n}}}"></span>, any normal distribution with mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="\mu "></span> and variance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="\sigma ^{2}"></span> is the distribution of the sum of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{n}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtext>n</mtext>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\text{n}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eaaf79c97c17e3d4d0a55bc13e965bacfbff279e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:1.676ex;" alt="{\displaystyle {\text{n}}}"></span> independent normal deviates, each with mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mu }{n}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>&#x03BC;<!-- μ --></mi>
<mi>n</mi>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {\mu }{n}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2d0f178e79529178aa38df33ff5eafe2ff05200" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.238ex; height:4.843ex;" alt="{\displaystyle {\frac {\mu }{n}}}"></span> and variance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\sigma ^{2}}{n}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>n</mi>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {\sigma ^{2}}{n}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/266591f45fcd859ccb660355c7c428b6af6737da" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.221ex; height:5.676ex;" alt="{\displaystyle {\frac {\sigma ^{2}}{n}}}"></span>. This property is called <a href="/wiki/Infinite_divisibility_(probability)" title="Infinite divisibility (probability)">infinite divisibility</a>.<sup id="cite_ref-42" class="reference"><a href="#cite_note-42">&#91;42&#93;</a></sup>
</p><p>Conversely, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{1}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f70b2694445a5901b24338a2e7a7e58f02a72a32" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="X_{1}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ad47c14b8a092f182512e76c96638aea6e3bea1" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="X_{2}"></span> are independent random variables and their sum <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}+X_{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{1}+X_{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f96d604ba472d9c8d3964bfb198de16b68e5f8a3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.797ex; height:2.509ex;" alt="X_{1}+X_{2}"></span> has a normal distribution, then both <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{1}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f70b2694445a5901b24338a2e7a7e58f02a72a32" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="X_{1}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ad47c14b8a092f182512e76c96638aea6e3bea1" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="X_{2}"></span> must be normal deviates.<sup id="cite_ref-43" class="reference"><a href="#cite_note-43">&#91;43&#93;</a></sup>
</p><p>This result is known as <a href="/wiki/Cram%C3%A9r%27s_decomposition_theorem" title="Cramér&#39;s decomposition theorem">Cramér's decomposition theorem</a>, and is equivalent to saying that the <a href="/wiki/Convolution" title="Convolution">convolution</a> of two distributions is normal if and only if both are normal. Cramér's theorem implies that a linear combination of independent non-Gaussian variables will never have an exactly normal distribution, although it may approach it arbitrarily closely.<sup id="cite_ref-Bryc_1995_35_29-1" class="reference"><a href="#cite_note-Bryc_1995_35-29">&#91;29&#93;</a></sup>
</p>
<h3><span id="Bernstein.27s_theorem"></span><span class="mw-headline" id="Bernstein's_theorem">Bernstein's theorem</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=31" title="Edit section: Bernstein&#039;s theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
<p>Bernstein's theorem states that if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="X"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="Y"></span> are independent and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X+Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>+</mo>
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X+Y}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/191744cf9cddeff3ab2e750e22bcfce7766d355e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.594ex; height:2.343ex;" alt="X+Y"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X-Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>X</mi>
<mo>&#x2212;<!-- --></mo>
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X-Y}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08754e521ae59759cd7ed7dc8b5c73c8b931f16a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.594ex; height:2.343ex;" alt="X-Y"></span> are also independent, then both <i>X</i> and <i>Y</i> must necessarily have normal distributions.<sup id="cite_ref-LK_44-0" class="reference"><a href="#cite_note-LK-44">&#91;44&#93;</a></sup><sup id="cite_ref-45" class="reference"><a href="#cite_note-45">&#91;45&#93;</a></sup>
</p><p>More generally, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{1},\ldots ,X_{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>&#x2026;<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{1},\ldots ,X_{n}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac794f5521dcce89913085a6d566e7cdb615dbb0" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.299ex; height:2.509ex;" alt="X_1, \ldots, X_n"></span> are independent random variables, then two distinct linear combinations <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum {a_{k}X_{k}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mo>&#x2211;<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\textstyle \sum {a_{k}X_{k}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d62b3dc666b11bdb962170ace940a297ff8d9c7f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.172ex; height:2.843ex;" alt="{\textstyle \sum {a_{k}X_{k}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum {b_{k}X_{k}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mo>&#x2211;<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\textstyle \sum {b_{k}X_{k}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/701fc598b855acd656fb94b99cfc1696dd881016" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.94ex; height:2.843ex;" alt="{\textstyle \sum {b_{k}X_{k}}}"></span>will be independent if and only if all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{k}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33c25229c6989c235f9cbb7908331f6d01d0abfe" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.013ex; height:2.509ex;" alt="X_{k}"></span> are normal and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum {a_{k}b_{k}\sigma _{k}^{2}=0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mo>&#x2211;<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<msub>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>=</mo>
<mn>0</mn>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\textstyle \sum {a_{k}b_{k}\sigma _{k}^{2}=0}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cca7bca2de00b5be911848276eca1c52c8ce765" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.923ex; height:3.176ex;" alt="{\textstyle \sum {a_{k}b_{k}\sigma _{k}^{2}=0}}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{k}^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma _{k}^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/065426e4746772f367e2476d16fa02f11460a70d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.416ex; height:3.176ex;" alt="\sigma_k^2"></span> denotes the variance of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle X_{k}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33c25229c6989c235f9cbb7908331f6d01d0abfe" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.013ex; height:2.509ex;" alt="X_{k}"></span>.<sup id="cite_ref-LK_44-1" class="reference"><a href="#cite_note-LK-44">&#91;44&#93;</a></sup>
</p>
<h3><span class="mw-headline" id="Extensions">Extensions</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=32" title="Edit section: Extensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
<p>The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called <i>normal</i> or <i>Gaussian</i> laws, so a certain ambiguity in names exists.
</p>
<ul><li>The <a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">multivariate normal distribution</a> describes the Gaussian law in the <i>k</i>-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>. A vector <span class="nowrap"><i>X</i> ∈ <b>R</b><sup><i>k</i></sup></span> is multivariate-normally distributed if any linear combination of its components <span class="nowrap">Σ<span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"><i>k</i></sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"><i>j</i>=1</sub></span><i>a<sub>j</sub> X<sub>j</sub></i></span> has a (univariate) normal distribution. The variance of <i>X</i> is a <i>k×k</i> symmetric positive-definite matrix&#160;<i>V</i>. The multivariate normal distribution is a special case of the <a href="/wiki/Elliptical_distribution" title="Elliptical distribution">elliptical distributions</a>. As such, its iso-density loci in the <i>k</i> = 2 case are <a href="/wiki/Ellipse" title="Ellipse">ellipses</a> and in the case of arbitrary <i>k</i> are <a href="/wiki/Ellipsoid" title="Ellipsoid">ellipsoids</a>.</li>
<li><a href="/wiki/Rectified_Gaussian_distribution" title="Rectified Gaussian distribution">Rectified Gaussian distribution</a> a rectified version of normal distribution with all the negative elements reset to 0</li>
<li><a href="/wiki/Complex_normal_distribution" title="Complex normal distribution">Complex normal distribution</a> deals with the complex normal vectors. A complex vector <span class="nowrap"><i>X</i> ∈ <b>C</b><sup><i>k</i></sup></span> is said to be normal if both its real and imaginary components jointly possess a 2<i>k</i>-dimensional multivariate normal distribution. The variance-covariance structure of <i>X</i> is described by two matrices: the <i>variance</i> matrix&#160;Γ, and the <i>relation</i> matrix&#160;<i>C</i>.</li>
<li><a href="/wiki/Matrix_normal_distribution" title="Matrix normal distribution">Matrix normal distribution</a> describes the case of normally distributed matrices.</li>
<li><a href="/wiki/Gaussian_process" title="Gaussian process">Gaussian processes</a> are the normally distributed <a href="/wiki/Stochastic_process" title="Stochastic process">stochastic processes</a>. These can be viewed as elements of some infinite-dimensional <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a>&#160;<i>H</i>, and thus are the analogues of multivariate normal vectors for the case <span class="nowrap"><i>k</i> = ∞</span>. A random element <span class="nowrap"><i>h</i> ∈ <i>H</i></span> is said to be normal if for any constant <span class="nowrap"><i>a</i> ∈ <i>H</i></span> the <a href="/wiki/Scalar_product" class="mw-redirect" title="Scalar product">scalar product</a> <span class="nowrap">(<i>a</i>, <i>h</i>)</span> has a (univariate) normal distribution. The variance structure of such Gaussian random element can be described in terms of the linear <i>covariance <span class="nowrap">operator K: H → H</span></i>. Several Gaussian processes became popular enough to have their own names:
<ul><li><a href="/wiki/Wiener_process" title="Wiener process">Brownian motion</a>,</li>
<li><a href="/wiki/Brownian_bridge" title="Brownian bridge">Brownian bridge</a>,</li>
<li><a href="/wiki/Ornstein%E2%80%93Uhlenbeck_process" title="OrnsteinUhlenbeck process">OrnsteinUhlenbeck process</a>.</li></ul></li>
<li><a href="/wiki/Gaussian_q-distribution" title="Gaussian q-distribution">Gaussian q-distribution</a> is an abstract mathematical construction that represents a <a href="/wiki/Q-analogue" class="mw-redirect" title="Q-analogue">q-analogue</a> of the normal distribution.</li>
<li>the <a href="/wiki/Q-Gaussian" class="mw-redirect" title="Q-Gaussian">q-Gaussian</a> is an analogue of the Gaussian distribution, in the sense that it maximises the <a href="/wiki/Tsallis_entropy" title="Tsallis entropy">Tsallis entropy</a>, and is one type of <a href="/wiki/Tsallis_distribution" title="Tsallis distribution">Tsallis distribution</a>. This distribution is different from the <a href="/wiki/Gaussian_q-distribution" title="Gaussian q-distribution">Gaussian q-distribution</a> above.</li>
<li>The <a href="/wiki/Kaniadakis_Gaussian_distribution" title="Kaniadakis Gaussian distribution">Kaniadakis <i>κ</i>-Gaussian distribution</a> is a generalization of the Gaussian distribution which arises from the <a href="/wiki/Kaniadakis_statistics" title="Kaniadakis statistics">Kaniadakis statistics</a>, being one of the <a href="/wiki/Kaniadakis_distribution" title="Kaniadakis distribution">Kaniadakis distributions</a>.</li></ul>
<p>A random variable <i>X</i> has a two-piece normal distribution if it has a distribution
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{X}(x)=N(\mu ,\sigma _{1}^{2}){\text{ if }}x\leq \mu }">
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<annotation encoding="application/x-tex">{\displaystyle f_{X}(x)=N(\mu ,\sigma _{1}^{2}){\text{ if }}x\leq \mu }</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6eb9de0278ee18288eb1a3d7cbe2f397a30ce29" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.052ex; height:3.176ex;" alt="{\displaystyle f_{X}(x)=N(\mu ,\sigma _{1}^{2}){\text{ if }}x\leq \mu }"></span></dd>
<dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{X}(x)=N(\mu ,\sigma _{2}^{2}){\text{ if }}x\geq \mu }">
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<annotation encoding="application/x-tex">{\displaystyle f_{X}(x)=N(\mu ,\sigma _{2}^{2}){\text{ if }}x\geq \mu }</annotation>
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<p>where <i>μ</i> is the mean and <i>σ</i><sub>1</sub> and <i>σ</i><sub>2</sub> are the standard deviations of the distribution to the left and right of the mean respectively.
</p><p>The mean, variance and third central moment of this distribution have been determined<sup id="cite_ref-John1982_46-0" class="reference"><a href="#cite_note-John1982-46">&#91;46&#93;</a></sup>
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {E} (X)=\mu +{\sqrt {\frac {2}{\pi }}}(\sigma _{2}-\sigma _{1})}">
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<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} (X)=\mu +{\sqrt {\frac {2}{\pi }}}(\sigma _{2}-\sigma _{1})}</annotation>
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<dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {V} (X)=\left(1-{\frac {2}{\pi }}\right)(\sigma _{2}-\sigma _{1})^{2}+\sigma _{1}\sigma _{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
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<mo>&#x2061;<!-- --></mo>
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<annotation encoding="application/x-tex">{\displaystyle \operatorname {V} (X)=\left(1-{\frac {2}{\pi }}\right)(\sigma _{2}-\sigma _{1})^{2}+\sigma _{1}\sigma _{2}}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec4a8615b5145dfc91cd114e8fb961d90ccecbda" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.681ex; height:6.176ex;" alt="{\displaystyle \operatorname {V} (X)=\left(1-{\frac {2}{\pi }}\right)(\sigma _{2}-\sigma _{1})^{2}+\sigma _{1}\sigma _{2}}"></span></dd>
<dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {T} (X)={\sqrt {\frac {2}{\pi }}}(\sigma _{2}-\sigma _{1})\left[\left({\frac {4}{\pi }}-1\right)(\sigma _{2}-\sigma _{1})^{2}+\sigma _{1}\sigma _{2}\right]}">
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<annotation encoding="application/x-tex">{\displaystyle \operatorname {T} (X)={\sqrt {\frac {2}{\pi }}}(\sigma _{2}-\sigma _{1})\left[\left({\frac {4}{\pi }}-1\right)(\sigma _{2}-\sigma _{1})^{2}+\sigma _{1}\sigma _{2}\right]}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9959f2c5186e2ed76884054edaf837a602ac6fac" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:53.363ex; height:6.343ex;" alt="{\displaystyle \operatorname {T} (X)={\sqrt {\frac {2}{\pi }}}(\sigma _{2}-\sigma _{1})\left[\left({\frac {4}{\pi }}-1\right)(\sigma _{2}-\sigma _{1})^{2}+\sigma _{1}\sigma _{2}\right]}"></span></dd></dl>
<p>where E(<i>X</i>), V(<i>X</i>) and T(<i>X</i>) are the mean, variance, and third central moment respectively.
</p><p>One of the main practical uses of the Gaussian law is to model the empirical distributions of many different random variables encountered in practice. In such case a possible extension would be a richer family of distributions, having more than two parameters and therefore being able to fit the empirical distribution more accurately. The examples of such extensions are:
</p>
<ul><li><a href="/wiki/Pearson_distribution" title="Pearson distribution">Pearson distribution</a> — a four-parameter family of probability distributions that extend the normal law to include different skewness and kurtosis values.</li>
<li>The <a href="/wiki/Generalized_normal_distribution" title="Generalized normal distribution">generalized normal distribution</a>, also known as the exponential power distribution, allows for distribution tails with thicker or thinner asymptotic behaviors.</li></ul>
<h2><span class="mw-headline" id="Statistical_inference">Statistical inference</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=33" title="Edit section: Statistical inference"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h2>
<h3><span class="mw-headline" id="Estimation_of_parameters">Estimation of parameters</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=34" title="Edit section: Estimation of parameters"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1033289096"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Maximum_likelihood#Continuous_distribution,_continuous_parameter_space" class="mw-redirect" title="Maximum likelihood">Maximum likelihood §&#160;Continuous distribution, continuous parameter space</a>; and <a href="/wiki/Gaussian_function#Estimation_of_parameters" title="Gaussian function">Gaussian function §&#160;Estimation of parameters</a></div>
<p>It is often the case that we do not know the parameters of the normal distribution, but instead want to <a href="/wiki/Estimation_theory" title="Estimation theory">estimate</a> them. That is, having a sample <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},\ldots ,x_{n})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<semantics>
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<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}(\mu ,\sigma ^{2})}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/863304aaa42a945f2f07d79facc3d2eebc845ce7" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.062ex; width:8.966ex; height:3.176ex;" alt="{\mathcal {N}}(\mu ,\sigma ^{2})"></span> population we would like to learn the approximate values of parameters <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="\mu "></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>&#x03C3;<!-- σ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="\sigma ^{2}"></span>. The standard approach to this problem is the <a href="/wiki/Maximum_likelihood" class="mw-redirect" title="Maximum likelihood">maximum likelihood</a> method, which requires maximization of the <i><a href="/wiki/Log-likelihood_function" class="mw-redirect" title="Log-likelihood function">log-likelihood function</a></i>:<span class="anchor" id="Log-likelihood"></span>
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln {\mathcal {L}}(\mu ,\sigma ^{2})=\sum _{i=1}^{n}\ln f(x_{i}\mid \mu ,\sigma ^{2})=-{\frac {n}{2}}\ln(2\pi )-{\frac {n}{2}}\ln \sigma ^{2}-{\frac {1}{2\sigma ^{2}}}\sum _{i=1}^{n}(x_{i}-\mu )^{2}.}">
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<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>&#x2223;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>,</mo>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>n</mi>
<mn>2</mn>
</mfrac>
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<mi>ln</mi>
<mo>&#x2061;<!-- --></mo>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mi>&#x03C0;<!-- π --></mi>
<mo stretchy="false">)</mo>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>n</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mi>ln</mi>
<mo>&#x2061;<!-- --></mo>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<munderover>
<mo>&#x2211;<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munderover>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \ln {\mathcal {L}}(\mu ,\sigma ^{2})=\sum _{i=1}^{n}\ln f(x_{i}\mid \mu ,\sigma ^{2})=-{\frac {n}{2}}\ln(2\pi )-{\frac {n}{2}}\ln \sigma ^{2}-{\frac {1}{2\sigma ^{2}}}\sum _{i=1}^{n}(x_{i}-\mu )^{2}.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/003faa08d27475dd2b029e9f7f0cebab17c0e147" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:76.486ex; height:6.843ex;" alt="{\displaystyle \ln {\mathcal {L}}(\mu ,\sigma ^{2})=\sum _{i=1}^{n}\ln f(x_{i}\mid \mu ,\sigma ^{2})=-{\frac {n}{2}}\ln(2\pi )-{\frac {n}{2}}\ln \sigma ^{2}-{\frac {1}{2\sigma ^{2}}}\sum _{i=1}^{n}(x_{i}-\mu )^{2}.}"></span></dd></dl>
<p>Taking derivatives with respect to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="\mu "></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="\sigma ^{2}"></span> and solving the resulting system of first order conditions yields the <i>maximum likelihood estimates</i>:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mu }}={\overline {x}}\equiv {\frac {1}{n}}\sum _{i=1}^{n}x_{i},\qquad {\hat {\sigma }}^{2}={\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>&#x03BC;<!-- μ --></mi>
<mo stretchy="false">&#x005E;<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo accent="false">&#x00AF;<!-- ¯ --></mo>
</mover>
</mrow>
<mo>&#x2261;<!-- ≡ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mi>n</mi>
</mfrac>
</mrow>
<munderover>
<mo>&#x2211;<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munderover>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>,</mo>
<mspace width="2em" />
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>&#x03C3;<!-- σ --></mi>
<mo stretchy="false">&#x005E;<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mi>n</mi>
</mfrac>
</mrow>
<munderover>
<mo>&#x2211;<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munderover>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo accent="false">&#x00AF;<!-- ¯ --></mo>
</mover>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\hat {\mu }}={\overline {x}}\equiv {\frac {1}{n}}\sum _{i=1}^{n}x_{i},\qquad {\hat {\sigma }}^{2}={\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0269b28e095780b5f1f76c94505841fbe51aeec2" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:44.592ex; height:6.843ex;" alt="{\hat {\mu }}={\overline {x}}\equiv {\frac {1}{n}}\sum _{i=1}^{n}x_{i},\qquad {\hat {\sigma }}^{2}={\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}."></span></dd></dl>
<p>Then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln {\mathcal {L}}({\hat {\mu }},{\hat {\sigma }}^{2})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ln</mi>
<mo>&#x2061;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">L</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>&#x03BC;<!-- μ --></mi>
<mo stretchy="false">&#x005E;<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mo>,</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>&#x03C3;<!-- σ --></mi>
<mo stretchy="false">&#x005E;<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \ln {\mathcal {L}}({\hat {\mu }},{\hat {\sigma }}^{2})}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4070a1dd43f598633edd0bc2a5dd8040f6670f1" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.559ex; height:3.176ex;" alt="{\displaystyle \ln {\mathcal {L}}({\hat {\mu }},{\hat {\sigma }}^{2})}"></span> is as follows:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln {\mathcal {L}}({\hat {\mu }},{\hat {\sigma }}^{2})=(-n/2)[log(2\pi {\hat {\sigma }}^{2})+1]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ln</mi>
<mo>&#x2061;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">L</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>&#x03BC;<!-- μ --></mi>
<mo stretchy="false">&#x005E;<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mo>,</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>&#x03C3;<!-- σ --></mi>
<mo stretchy="false">&#x005E;<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mo>&#x2212;<!-- --></mo>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">[</mo>
<mi>l</mi>
<mi>o</mi>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mi>&#x03C0;<!-- π --></mi>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>&#x03C3;<!-- σ --></mi>
<mo stretchy="false">&#x005E;<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \ln {\mathcal {L}}({\hat {\mu }},{\hat {\sigma }}^{2})=(-n/2)[log(2\pi {\hat {\sigma }}^{2})+1]}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3300a94c4ca3bf6091b63eba6109700e88aa101a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.916ex; height:3.176ex;" alt="{\displaystyle \ln {\mathcal {L}}({\hat {\mu }},{\hat {\sigma }}^{2})=(-n/2)[log(2\pi {\hat {\sigma }}^{2})+1]}"></span></dd></dl>
<h4><span class="mw-headline" id="Sample_mean">Sample mean</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=35" title="Edit section: Sample mean"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h4>
<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1033289096"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Standard_error_of_the_mean" class="mw-redirect" title="Standard error of the mean">Standard error of the mean</a></div>
<p>Estimator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\hat {\mu }}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mstyle displaystyle="false" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>&#x03BC;<!-- μ --></mi>
<mo stretchy="false">&#x005E;<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \textstyle {\hat {\mu }}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adefcea6c129cd8f06e8fc941a5f760cb9c4d5b4" class="mwe-math-fallback-image-inline mw-invert" style="vertical-align: -0.838ex; width:1.402ex; height:2.676ex;" aria-hidden="true" alt="{\displaystyle \textstyle {\hat {\mu }}}"></span> is called the <i><a href="/wiki/Sample_mean" class="mw-redirect" title="Sample mean">sample mean</a></i>, since it is the arithmetic mean of all observations. The statistic <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\overline {x}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mstyle displaystyle="false" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo accent="false">&#x00AF;<!-- ¯ --></mo>
</mover>
</mrow>
</mstyle>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \textstyle {\overline {x}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c74eb776989f75b04948837080faa9ebc08c8cd3" class="mwe-math-fallback-image-inline mw-invert" style="vertical-align: -0.338ex; width:1.445ex; height:2.343ex;" aria-hidden="true" alt="{\displaystyle \textstyle {\overline {x}}}"></span> is <a href="/wiki/Complete_statistic" class="mw-redirect" title="Complete statistic">complete</a> and <a href="/wiki/Sufficient_statistic" title="Sufficient statistic">sufficient</a> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="\mu "></span>, and therefore by the <a href="/wiki/Lehmann%E2%80%93Scheff%C3%A9_theorem" title="LehmannScheffé theorem">LehmannScheffé theorem</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\hat {\mu }}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mstyle displaystyle="false" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>&#x03BC;<!-- μ --></mi>
<mo stretchy="false">&#x005E;<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \textstyle {\hat {\mu }}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adefcea6c129cd8f06e8fc941a5f760cb9c4d5b4" class="mwe-math-fallback-image-inline mw-invert" style="vertical-align: -0.838ex; width:1.402ex; height:2.676ex;" aria-hidden="true" alt="{\displaystyle \textstyle {\hat {\mu }}}"></span> is the <a href="/wiki/Uniformly_minimum_variance_unbiased" class="mw-redirect" title="Uniformly minimum variance unbiased">uniformly minimum variance unbiased</a> (UMVU) estimator.<sup id="cite_ref-Kri127_47-0" class="reference"><a href="#cite_note-Kri127-47">&#91;47&#93;</a></sup> In finite samples it is distributed normally:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mu }}\sim {\mathcal {N}}(\mu ,\sigma ^{2}/n).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>&#x03BC;<!-- μ --></mi>
<mo stretchy="false">&#x005E;<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mo>&#x223C;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">N</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>,</mo>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\hat {\mu }}\sim {\mathcal {N}}(\mu ,\sigma ^{2}/n).}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8f1fbb023c73b0f4010814107ac36419b16a226" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.609ex; height:3.176ex;" alt="{\displaystyle {\hat {\mu }}\sim {\mathcal {N}}(\mu ,\sigma ^{2}/n).}"></span></dd></dl>
<p>The variance of this estimator is equal to the <i>μμ</i>-element of the inverse <a href="/wiki/Fisher_information_matrix" class="mw-redirect" title="Fisher information matrix">Fisher information matrix</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\mathcal {I}}^{-1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mstyle displaystyle="false" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">I</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
</mrow>
</msup>
</mstyle>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \textstyle {\mathcal {I}}^{-1}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98cf99dd702e8c61031251ee2506b639a6eff98f" class="mwe-math-fallback-image-inline mw-invert" style="vertical-align: -0.338ex; margin-left: -0.069ex; width:3.961ex; height:2.676ex;" aria-hidden="true" alt="{\displaystyle \textstyle {\mathcal {I}}^{-1}}"></span>. This implies that the estimator is <a href="/wiki/Efficient_estimator" class="mw-redirect" title="Efficient estimator">finite-sample efficient</a>. Of practical importance is the fact that the <a href="/wiki/Standard_error_(statistics)" class="mw-redirect" title="Standard error (statistics)">standard error</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\hat {\mu }}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mstyle displaystyle="false" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>&#x03BC;<!-- μ --></mi>
<mo stretchy="false">&#x005E;<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \textstyle {\hat {\mu }}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adefcea6c129cd8f06e8fc941a5f760cb9c4d5b4" class="mwe-math-fallback-image-inline mw-invert" style="vertical-align: -0.838ex; width:1.402ex; height:2.676ex;" aria-hidden="true" alt="{\displaystyle \textstyle {\hat {\mu }}}"></span> is proportional to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle 1/{\sqrt {n}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mstyle displaystyle="false" scriptlevel="0">
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>n</mi>
</msqrt>
</mrow>
</mstyle>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \textstyle 1/{\sqrt {n}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cd0024843448c587ee8246c08fe5af7fb03cc95" class="mwe-math-fallback-image-inline mw-invert" style="vertical-align: -0.838ex; width:5.656ex; height:2.843ex;" aria-hidden="true" alt="{\displaystyle \textstyle 1/{\sqrt {n}}}"></span>, that is, if one wishes to decrease the standard error by a factor of 10, one must increase the number of points in the sample by a factor of 100. This fact is widely used in determining sample sizes for opinion polls and the number of trials in <a href="/wiki/Monte_Carlo_simulation" class="mw-redirect" title="Monte Carlo simulation">Monte Carlo simulations</a>.
</p><p>From the standpoint of the <a href="/wiki/Asymptotic_theory_(statistics)" title="Asymptotic theory (statistics)">asymptotic theory</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\hat {\mu }}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mstyle displaystyle="false" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>&#x03BC;<!-- μ --></mi>
<mo stretchy="false">&#x005E;<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \textstyle {\hat {\mu }}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adefcea6c129cd8f06e8fc941a5f760cb9c4d5b4" class="mwe-math-fallback-image-inline mw-invert" style="vertical-align: -0.838ex; width:1.402ex; height:2.676ex;" aria-hidden="true" alt="{\displaystyle \textstyle {\hat {\mu }}}"></span> is <a href="/wiki/Consistent_estimator" title="Consistent estimator">consistent</a>, that is, it <a href="/wiki/Convergence_in_probability" class="mw-redirect" title="Convergence in probability">converges in probability</a> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="\mu "></span> as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\rightarrow \infty }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo stretchy="false">&#x2192;<!-- → --></mo>
<mi mathvariant="normal">&#x221E;<!-- ∞ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n\rightarrow \infty }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9702f04f2d0e5b887b99faeeffb0c4cfd8263eee" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.333ex; height:1.843ex;" alt="n\rightarrow \infty "></span>. The estimator is also <a href="/wiki/Asymptotic_normality" class="mw-redirect" title="Asymptotic normality">asymptotically normal</a>, which is a simple corollary of the fact that it is normal in finite samples:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {n}}({\hat {\mu }}-\mu )\,{\xrightarrow {d}}\,{\mathcal {N}}(0,\sigma ^{2}).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>n</mi>
</msqrt>
</mrow>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>&#x03BC;<!-- μ --></mi>
<mo stretchy="false">&#x005E;<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace" />
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mo>&#x2192;</mo>
<mpadded width="+0.611em" lspace="0.278em" voffset=".15em">
<mi>d</mi>
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</mover>
</mrow>
<mspace width="thinmathspace" />
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">N</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>,</mo>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\sqrt {n}}({\hat {\mu }}-\mu )\,{\xrightarrow {d}}\,{\mathcal {N}}(0,\sigma ^{2}).}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c24f59324cc7b062e4996a0bd754ceb7493355a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-top: -0.311ex; width:23.194ex; height:4.343ex;" alt="{\displaystyle {\sqrt {n}}({\hat {\mu }}-\mu )\,{\xrightarrow {d}}\,{\mathcal {N}}(0,\sigma ^{2}).}"></span></dd></dl>
<h4><span class="mw-headline" id="Sample_variance">Sample variance</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=36" title="Edit section: Sample variance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h4>
<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1033289096"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Standard_deviation#Estimation" title="Standard deviation">Standard deviation §&#160;Estimation</a>, and <a href="/wiki/Variance#Estimation" title="Variance">Variance §&#160;Estimation</a></div>
<p>The estimator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\hat {\sigma }}^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mstyle displaystyle="false" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>&#x03C3;<!-- σ --></mi>
<mo stretchy="false">&#x005E;<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \textstyle {\hat {\sigma }}^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dbeb6ca1eacf73ca838981e36035f66f8449084" class="mwe-math-fallback-image-inline mw-invert" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" aria-hidden="true" alt="{\displaystyle \textstyle {\hat {\sigma }}^{2}}"></span> is called the <i><a href="/wiki/Sample_variance" class="mw-redirect" title="Sample variance">sample variance</a></i>, since it is the variance of the sample (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},\ldots ,x_{n})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>&#x2026;<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (x_{1},\ldots ,x_{n})}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7935f7983d8a5ae59fea84efe65415235fa7c47b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.92ex; height:2.843ex;" alt="(x_1, \ldots, x_n)"></span>). In practice, another estimator is often used instead of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\hat {\sigma }}^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mstyle displaystyle="false" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>&#x03C3;<!-- σ --></mi>
<mo stretchy="false">&#x005E;<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \textstyle {\hat {\sigma }}^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dbeb6ca1eacf73ca838981e36035f66f8449084" class="mwe-math-fallback-image-inline mw-invert" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" aria-hidden="true" alt="{\displaystyle \textstyle {\hat {\sigma }}^{2}}"></span>. This other estimator is denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>s</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58d7841cee3671436949ee789b84a848fd150bd3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.145ex; height:2.676ex;" alt="s^{2}"></span>, and is also called the <i>sample variance</i>, which represents a certain ambiguity in terminology; its square root <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>s</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="s"></span> is called the <i>sample standard deviation</i>. The estimator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>s</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58d7841cee3671436949ee789b84a848fd150bd3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.145ex; height:2.676ex;" alt="s^{2}"></span> differs from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\hat {\sigma }}^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mstyle displaystyle="false" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>&#x03C3;<!-- σ --></mi>
<mo stretchy="false">&#x005E;<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \textstyle {\hat {\sigma }}^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dbeb6ca1eacf73ca838981e36035f66f8449084" class="mwe-math-fallback-image-inline mw-invert" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" aria-hidden="true" alt="{\displaystyle \textstyle {\hat {\sigma }}^{2}}"></span> by having <span class="nowrap">(<i>n</i> 1)</span> instead of&#160;<i>n</i> in the denominator (the so-called <a href="/wiki/Bessel%27s_correction" title="Bessel&#39;s correction">Bessel's correction</a>):
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s^{2}={\frac {n}{n-1}}{\hat {\sigma }}^{2}={\frac {1}{n-1}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>s</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>n</mi>
<mrow>
<mi>n</mi>
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
</mrow>
</mfrac>
</mrow>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>&#x03C3;<!-- σ --></mi>
<mo stretchy="false">&#x005E;<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mi>n</mi>
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
</mrow>
</mfrac>
</mrow>
<munderover>
<mo>&#x2211;<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munderover>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo accent="false">&#x00AF;<!-- ¯ --></mo>
</mover>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s^{2}={\frac {n}{n-1}}{\hat {\sigma }}^{2}={\frac {1}{n-1}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb09766b1fa03887c9ec7f7254e3b25f94224532" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:36.86ex; height:6.843ex;" alt="{\displaystyle s^{2}={\frac {n}{n-1}}{\hat {\sigma }}^{2}={\frac {1}{n-1}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}.}"></span></dd></dl>
<p>The difference between <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>s</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58d7841cee3671436949ee789b84a848fd150bd3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.145ex; height:2.676ex;" alt="s^{2}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\hat {\sigma }}^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mstyle displaystyle="false" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>&#x03C3;<!-- σ --></mi>
<mo stretchy="false">&#x005E;<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \textstyle {\hat {\sigma }}^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dbeb6ca1eacf73ca838981e36035f66f8449084" class="mwe-math-fallback-image-inline mw-invert" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" aria-hidden="true" alt="{\displaystyle \textstyle {\hat {\sigma }}^{2}}"></span> becomes negligibly small for large <i>n</i><span class="nowrap" style="padding-left:0.1em;">&#39;</span>s. In finite samples however, the motivation behind the use of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>s</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58d7841cee3671436949ee789b84a848fd150bd3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.145ex; height:2.676ex;" alt="s^{2}"></span> is that it is an <a href="/wiki/Unbiased_estimator" class="mw-redirect" title="Unbiased estimator">unbiased estimator</a> of the underlying parameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="\sigma ^{2}"></span>, whereas <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\hat {\sigma }}^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mstyle displaystyle="false" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>&#x03C3;<!-- σ --></mi>
<mo stretchy="false">&#x005E;<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \textstyle {\hat {\sigma }}^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dbeb6ca1eacf73ca838981e36035f66f8449084" class="mwe-math-fallback-image-inline mw-invert" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" aria-hidden="true" alt="{\displaystyle \textstyle {\hat {\sigma }}^{2}}"></span> is biased. Also, by the LehmannScheffé theorem the estimator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>s</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58d7841cee3671436949ee789b84a848fd150bd3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.145ex; height:2.676ex;" alt="s^{2}"></span> is uniformly minimum variance unbiased (<a href="/wiki/Minimum-variance_unbiased_estimator" title="Minimum-variance unbiased estimator">UMVU</a>),<sup id="cite_ref-Kri127_47-1" class="reference"><a href="#cite_note-Kri127-47">&#91;47&#93;</a></sup> which makes it the "best" estimator among all unbiased ones. However it can be shown that the biased estimator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\hat {\sigma }}^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mstyle displaystyle="false" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>&#x03C3;<!-- σ --></mi>
<mo stretchy="false">&#x005E;<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \textstyle {\hat {\sigma }}^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dbeb6ca1eacf73ca838981e36035f66f8449084" class="mwe-math-fallback-image-inline mw-invert" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" aria-hidden="true" alt="{\displaystyle \textstyle {\hat {\sigma }}^{2}}"></span> is better than the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>s</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58d7841cee3671436949ee789b84a848fd150bd3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.145ex; height:2.676ex;" alt="s^{2}"></span> in terms of the <a href="/wiki/Mean_squared_error" title="Mean squared error">mean squared error</a> (MSE) criterion. In finite samples both <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>s</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58d7841cee3671436949ee789b84a848fd150bd3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.145ex; height:2.676ex;" alt="s^{2}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\hat {\sigma }}^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mstyle displaystyle="false" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>&#x03C3;<!-- σ --></mi>
<mo stretchy="false">&#x005E;<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \textstyle {\hat {\sigma }}^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dbeb6ca1eacf73ca838981e36035f66f8449084" class="mwe-math-fallback-image-inline mw-invert" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" aria-hidden="true" alt="{\displaystyle \textstyle {\hat {\sigma }}^{2}}"></span> have scaled <a href="/wiki/Chi-squared_distribution" title="Chi-squared distribution">chi-squared distribution</a> with <span class="nowrap">(<i>n</i> 1)</span> degrees of freedom:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s^{2}\sim {\frac {\sigma ^{2}}{n-1}}\cdot \chi _{n-1}^{2},\qquad {\hat {\sigma }}^{2}\sim {\frac {\sigma ^{2}}{n}}\cdot \chi _{n-1}^{2}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>s</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>&#x223C;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mrow>
<mi>n</mi>
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
</mrow>
</mfrac>
</mrow>
<mo>&#x22C5;<!-- ⋅ --></mo>
<msubsup>
<mi>&#x03C7;<!-- χ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>,</mo>
<mspace width="2em" />
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>&#x03C3;<!-- σ --></mi>
<mo stretchy="false">&#x005E;<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>&#x223C;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mi>n</mi>
</mfrac>
</mrow>
<mo>&#x22C5;<!-- ⋅ --></mo>
<msubsup>
<mi>&#x03C7;<!-- χ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s^{2}\sim {\frac {\sigma ^{2}}{n-1}}\cdot \chi _{n-1}^{2},\qquad {\hat {\sigma }}^{2}\sim {\frac {\sigma ^{2}}{n}}\cdot \chi _{n-1}^{2}.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b55e6d2c748d5ba1ff42692650492b9506ab164d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:39.412ex; height:5.843ex;" alt="{\displaystyle s^{2}\sim {\frac {\sigma ^{2}}{n-1}}\cdot \chi _{n-1}^{2},\qquad {\hat {\sigma }}^{2}\sim {\frac {\sigma ^{2}}{n}}\cdot \chi _{n-1}^{2}.}"></span></dd></dl>
<p>The first of these expressions shows that the variance of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>s</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58d7841cee3671436949ee789b84a848fd150bd3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.145ex; height:2.676ex;" alt="s^{2}"></span> is equal to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\sigma ^{4}/(n-1)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>2</mn>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 2\sigma ^{4}/(n-1)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc5d54f356da083e73a2d171a278ff7d9082a085" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.917ex; height:3.176ex;" alt="{\displaystyle 2\sigma ^{4}/(n-1)}"></span>, which is slightly greater than the <i>σσ</i>-element of the inverse Fisher information matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\mathcal {I}}^{-1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mstyle displaystyle="false" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">I</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
</mrow>
</msup>
</mstyle>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \textstyle {\mathcal {I}}^{-1}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98cf99dd702e8c61031251ee2506b639a6eff98f" class="mwe-math-fallback-image-inline mw-invert" style="vertical-align: -0.338ex; margin-left: -0.069ex; width:3.961ex; height:2.676ex;" aria-hidden="true" alt="{\displaystyle \textstyle {\mathcal {I}}^{-1}}"></span>. Thus, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>s</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58d7841cee3671436949ee789b84a848fd150bd3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.145ex; height:2.676ex;" alt="s^{2}"></span> is not an efficient estimator for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="\sigma ^{2}"></span>, and moreover, since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>s</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58d7841cee3671436949ee789b84a848fd150bd3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.145ex; height:2.676ex;" alt="s^{2}"></span> is UMVU, we can conclude that the finite-sample efficient estimator for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="\sigma ^{2}"></span> does not exist.
</p><p>Applying the asymptotic theory, both estimators <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>s</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle s^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58d7841cee3671436949ee789b84a848fd150bd3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.145ex; height:2.676ex;" alt="s^{2}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\hat {\sigma }}^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mstyle displaystyle="false" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>&#x03C3;<!-- σ --></mi>
<mo stretchy="false">&#x005E;<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \textstyle {\hat {\sigma }}^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dbeb6ca1eacf73ca838981e36035f66f8449084" class="mwe-math-fallback-image-inline mw-invert" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" aria-hidden="true" alt="{\displaystyle \textstyle {\hat {\sigma }}^{2}}"></span> are consistent, that is they converge in probability to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="\sigma ^{2}"></span> as the sample size <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\rightarrow \infty }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo stretchy="false">&#x2192;<!-- → --></mo>
<mi mathvariant="normal">&#x221E;<!-- ∞ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n\rightarrow \infty }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9702f04f2d0e5b887b99faeeffb0c4cfd8263eee" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.333ex; height:1.843ex;" alt="n\rightarrow \infty "></span>. The two estimators are also both asymptotically normal:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {n}}({\hat {\sigma }}^{2}-\sigma ^{2})\simeq {\sqrt {n}}(s^{2}-\sigma ^{2})\,{\xrightarrow {d}}\,{\mathcal {N}}(0,2\sigma ^{4}).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>n</mi>
</msqrt>
</mrow>
<mo stretchy="false">(</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>&#x03C3;<!-- σ --></mi>
<mo stretchy="false">&#x005E;<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>&#x2212;<!-- --></mo>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>&#x2243;<!-- ≃ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>n</mi>
</msqrt>
</mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>s</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>&#x2212;<!-- --></mo>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace" />
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mo>&#x2192;</mo>
<mpadded width="+0.611em" lspace="0.278em" voffset=".15em">
<mi>d</mi>
</mpadded>
</mover>
</mrow>
<mspace width="thinmathspace" />
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">N</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>,</mo>
<mn>2</mn>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>4</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\sqrt {n}}({\hat {\sigma }}^{2}-\sigma ^{2})\simeq {\sqrt {n}}(s^{2}-\sigma ^{2})\,{\xrightarrow {d}}\,{\mathcal {N}}(0,2\sigma ^{4}).}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9fad7f0b05e93efdc409365219707701b64c33b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-top: -0.311ex; width:41.93ex; height:4.343ex;" alt="{\displaystyle {\sqrt {n}}({\hat {\sigma }}^{2}-\sigma ^{2})\simeq {\sqrt {n}}(s^{2}-\sigma ^{2})\,{\xrightarrow {d}}\,{\mathcal {N}}(0,2\sigma ^{4}).}"></span></dd></dl>
<p>In particular, both estimators are asymptotically efficient for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.385ex; height:2.676ex;" alt="\sigma ^{2}"></span>.
</p>
<h3><span class="mw-headline" id="Confidence_intervals">Confidence intervals</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=37" title="Edit section: Confidence intervals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1033289096"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Studentization" title="Studentization">Studentization</a> and <a href="/wiki/3-sigma_rule" class="mw-redirect" title="3-sigma rule">3-sigma rule</a></div>
<p>By <a href="/wiki/Cochran%27s_theorem" title="Cochran&#39;s theorem">Cochran's theorem</a>, for normal distributions the sample mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\hat {\mu }}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mstyle displaystyle="false" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>&#x03BC;<!-- μ --></mi>
<mo stretchy="false">&#x005E;<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \textstyle {\hat {\mu }}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adefcea6c129cd8f06e8fc941a5f760cb9c4d5b4" class="mwe-math-fallback-image-inline mw-invert" style="vertical-align: -0.838ex; width:1.402ex; height:2.676ex;" aria-hidden="true" alt="{\displaystyle \textstyle {\hat {\mu }}}"></span> and the sample variance <i>s</i><sup>2</sup> are <a href="/wiki/Independence_(probability_theory)" title="Independence (probability theory)">independent</a>, which means there can be no gain in considering their <a href="/wiki/Joint_distribution" class="mw-redirect" title="Joint distribution">joint distribution</a>. There is also a converse theorem: if in a sample the sample mean and sample variance are independent, then the sample must have come from the normal distribution. The independence between <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\hat {\mu }}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mstyle displaystyle="false" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>&#x03BC;<!-- μ --></mi>
<mo stretchy="false">&#x005E;<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \textstyle {\hat {\mu }}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adefcea6c129cd8f06e8fc941a5f760cb9c4d5b4" class="mwe-math-fallback-image-inline mw-invert" style="vertical-align: -0.838ex; width:1.402ex; height:2.676ex;" aria-hidden="true" alt="{\displaystyle \textstyle {\hat {\mu }}}"></span> and <i>s</i> can be employed to construct the so-called <i>t-statistic</i>:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t={\frac {{\hat {\mu }}-\mu }{s/{\sqrt {n}}}}={\frac {{\overline {x}}-\mu }{\sqrt {{\frac {1}{n(n-1)}}\sum (x_{i}-{\overline {x}})^{2}}}}\sim t_{n-1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>t</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>&#x03BC;<!-- μ --></mi>
<mo stretchy="false">&#x005E;<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
</mrow>
<mrow>
<mi>s</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>n</mi>
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</mfrac>
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<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo accent="false">&#x00AF;<!-- ¯ --></mo>
</mover>
</mrow>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
</mrow>
<msqrt>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mi>n</mi>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
<mo>&#x2211;<!-- ∑ --></mo>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo accent="false">&#x00AF;<!-- ¯ --></mo>
</mover>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</msqrt>
</mfrac>
</mrow>
<mo>&#x223C;<!-- --></mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
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</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle t={\frac {{\hat {\mu }}-\mu }{s/{\sqrt {n}}}}={\frac {{\overline {x}}-\mu }{\sqrt {{\frac {1}{n(n-1)}}\sum (x_{i}-{\overline {x}})^{2}}}}\sim t_{n-1}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f4ea0fbb1b9bdbcef271db64817c384d43497a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:42.241ex; height:8.509ex;" alt="{\displaystyle t={\frac {{\hat {\mu }}-\mu }{s/{\sqrt {n}}}}={\frac {{\overline {x}}-\mu }{\sqrt {{\frac {1}{n(n-1)}}\sum (x_{i}-{\overline {x}})^{2}}}}\sim t_{n-1}}"></span></dd></dl>
<p>This quantity <i>t</i> has the <a href="/wiki/Student%27s_t-distribution" title="Student&#39;s t-distribution">Student's t-distribution</a> with <span class="nowrap">(<i>n</i> 1)</span> degrees of freedom, and it is an <a href="/wiki/Ancillary_statistic" title="Ancillary statistic">ancillary statistic</a> (independent of the value of the parameters). Inverting the distribution of this <i>t</i>-statistics will allow us to construct the <a href="/wiki/Confidence_interval" title="Confidence interval">confidence interval</a> for <i>μ</i>;<sup id="cite_ref-48" class="reference"><a href="#cite_note-48">&#91;48&#93;</a></sup> similarly, inverting the <i>χ</i><sup>2</sup> distribution of the statistic <i>s</i><sup>2</sup> will give us the confidence interval for <i>σ</i><sup>2</sup>:<sup id="cite_ref-49" class="reference"><a href="#cite_note-49">&#91;49&#93;</a></sup>
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \in \left[{\hat {\mu }}-t_{n-1,1-\alpha /2}{\frac {1}{\sqrt {n}}}s,{\hat {\mu }}+t_{n-1,1-\alpha /2}{\frac {1}{\sqrt {n}}}s\right],}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
<mo>&#x2208;<!-- ∈ --></mo>
<mrow>
<mo>[</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>&#x03BC;<!-- μ --></mi>
<mo stretchy="false">&#x005E;<!-- ^ --></mo>
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</mrow>
</mrow>
<mo>&#x2212;<!-- --></mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
<mo>,</mo>
<mn>1</mn>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03B1;<!-- α --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
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<mi>s</mi>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
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<mover>
<mi>&#x03BC;<!-- μ --></mi>
<mo stretchy="false">&#x005E;<!-- ^ --></mo>
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<mo>+</mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
<mo>,</mo>
<mn>1</mn>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03B1;<!-- α --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<msqrt>
<mi>n</mi>
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</mrow>
<mi>s</mi>
</mrow>
<mo>]</mo>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu \in \left[{\hat {\mu }}-t_{n-1,1-\alpha /2}{\frac {1}{\sqrt {n}}}s,{\hat {\mu }}+t_{n-1,1-\alpha /2}{\frac {1}{\sqrt {n}}}s\right],}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6e3068587bfbaf61a549a39b518757119bfb846" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:46.588ex; height:6.509ex;" alt="{\displaystyle \mu \in \left[{\hat {\mu }}-t_{n-1,1-\alpha /2}{\frac {1}{\sqrt {n}}}s,{\hat {\mu }}+t_{n-1,1-\alpha /2}{\frac {1}{\sqrt {n}}}s\right],}"></span></dd>
<dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}\in \left[{\frac {(n-1)s^{2}}{\chi _{n-1,1-\alpha /2}^{2}}},{\frac {(n-1)s^{2}}{\chi _{n-1,\alpha /2}^{2}}}\right],}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>&#x2208;<!-- ∈ --></mo>
<mrow>
<mo>[</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<msup>
<mi>s</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<msubsup>
<mi>&#x03C7;<!-- χ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
<mo>,</mo>
<mn>1</mn>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03B1;<!-- α --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msubsup>
</mfrac>
</mrow>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<msup>
<mi>s</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<msubsup>
<mi>&#x03C7;<!-- χ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
<mo>,</mo>
<mi>&#x03B1;<!-- α --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msubsup>
</mfrac>
</mrow>
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<mo>]</mo>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}\in \left[{\frac {(n-1)s^{2}}{\chi _{n-1,1-\alpha /2}^{2}}},{\frac {(n-1)s^{2}}{\chi _{n-1,\alpha /2}^{2}}}\right],}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3549a31cb861d9e479c232271cb88b019cfa9fd5" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:31.055ex; height:7.843ex;" alt="{\displaystyle \sigma ^{2}\in \left[{\frac {(n-1)s^{2}}{\chi _{n-1,1-\alpha /2}^{2}}},{\frac {(n-1)s^{2}}{\chi _{n-1,\alpha /2}^{2}}}\right],}"></span></dd></dl>
<p>where <i>t<sub>k,p</sub></i> and <span style="white-space: nowrap;"><span style="font-style: italic;">χ</span><span style="font-size: 40%;">&#160;</span><span style="font-size: 70%;"><span style="display:inline-block; vertical-align: -0.4em; line-height:1.1em;">2<br /><i>k,p</i></span><span style="font-size: 40%;">&#160;</span></span></span> are the <i>p</i>th <a href="/wiki/Quantile" title="Quantile">quantiles</a> of the <i>t</i>- and <i>χ</i><sup>2</sup>-distributions respectively. These confidence intervals are of the <i><a href="/wiki/Confidence_level" class="mw-redirect" title="Confidence level">confidence level</a></i> <span class="nowrap">1 <i>α</i></span>, meaning that the true values <i>μ</i> and <i>σ</i><sup>2</sup> fall outside of these intervals with probability (or <a href="/wiki/Significance_level" class="mw-redirect" title="Significance level">significance level</a>) <i>α</i>. In practice people usually take <span class="nowrap"><i>α</i> = 5%</span>, resulting in the 95% confidence intervals.
</p><p>Approximate formulas can be derived from the asymptotic distributions of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\hat {\mu }}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mstyle displaystyle="false" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>&#x03BC;<!-- μ --></mi>
<mo stretchy="false">&#x005E;<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \textstyle {\hat {\mu }}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adefcea6c129cd8f06e8fc941a5f760cb9c4d5b4" class="mwe-math-fallback-image-inline mw-invert" style="vertical-align: -0.838ex; width:1.402ex; height:2.676ex;" aria-hidden="true" alt="{\displaystyle \textstyle {\hat {\mu }}}"></span> and <i>s</i><sup>2</sup>:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \in \left[{\hat {\mu }}-|z_{\alpha /2}|{\frac {1}{\sqrt {n}}}s,{\hat {\mu }}+|z_{\alpha /2}|{\frac {1}{\sqrt {n}}}s\right],}">
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<mi>&#x03BC;<!-- μ --></mi>
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<mo>&#x2212;<!-- --></mo>
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<mi>z</mi>
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<mi>&#x03B1;<!-- α --></mi>
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<mi>&#x03BC;<!-- μ --></mi>
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<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
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<mi>z</mi>
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<mi>&#x03B1;<!-- α --></mi>
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<mo>/</mo>
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<mo>,</mo>
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<annotation encoding="application/x-tex">{\displaystyle \mu \in \left[{\hat {\mu }}-|z_{\alpha /2}|{\frac {1}{\sqrt {n}}}s,{\hat {\mu }}+|z_{\alpha /2}|{\frac {1}{\sqrt {n}}}s\right],}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e13883f93c0a1405e71bd105685ecac6b4c84089" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:38.37ex; height:6.509ex;" alt="{\displaystyle \mu \in \left[{\hat {\mu }}-|z_{\alpha /2}|{\frac {1}{\sqrt {n}}}s,{\hat {\mu }}+|z_{\alpha /2}|{\frac {1}{\sqrt {n}}}s\right],}"></span></dd>
<dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{2}\in \left[s^{2}-|z_{\alpha /2}|{\frac {\sqrt {2}}{\sqrt {n}}}s^{2},s^{2}+|z_{\alpha /2}|{\frac {\sqrt {2}}{\sqrt {n}}}s^{2}\right],}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>&#x03C3;<!-- σ --></mi>
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<mn>2</mn>
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<mi>z</mi>
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<mi>z</mi>
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<mi>&#x03B1;<!-- α --></mi>
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<mo>/</mo>
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<annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}\in \left[s^{2}-|z_{\alpha /2}|{\frac {\sqrt {2}}{\sqrt {n}}}s^{2},s^{2}+|z_{\alpha /2}|{\frac {\sqrt {2}}{\sqrt {n}}}s^{2}\right],}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3e7c940eb3f6f50af62200ae75e10435ef8dfe6" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:42.947ex; height:6.843ex;" alt="{\displaystyle \sigma ^{2}\in \left[s^{2}-|z_{\alpha /2}|{\frac {\sqrt {2}}{\sqrt {n}}}s^{2},s^{2}+|z_{\alpha /2}|{\frac {\sqrt {2}}{\sqrt {n}}}s^{2}\right],}"></span></dd></dl>
<p>The approximate formulas become valid for large values of <i>n</i>, and are more convenient for the manual calculation since the standard normal quantiles <i>z</i><sub><i>α</i>/2</sub> do not depend on <i>n</i>. In particular, the most popular value of <span class="nowrap"><i>α</i> = 5%</span>, results in <span class="nowrap">|<i>z</i><sub>0.025</sub>| = <a href="/wiki/1.96" class="mw-redirect" title="1.96">1.96</a></span>.
</p>
<h3><span class="mw-headline" id="Normality_tests">Normality tests</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=38" title="Edit section: Normality tests"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1033289096"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Normality_tests" class="mw-redirect" title="Normality tests">Normality tests</a></div>
<p>Normality tests assess the likelihood that the given data set {<i>x</i><sub>1</sub>, ..., <i>x<sub>n</sub></i>} comes from a normal distribution. Typically the <a href="/wiki/Null_hypothesis" title="Null hypothesis">null hypothesis</a> <i>H</i><sub>0</sub> is that the observations are distributed normally with unspecified mean <i>μ</i> and variance <i>σ</i><sup>2</sup>, versus the alternative <i>H<sub>a</sub></i> that the distribution is arbitrary. Many tests (over 40) have been devised for this problem. The more prominent of them are outlined below:
</p><p><b>Diagnostic plots</b> are more intuitively appealing but subjective at the same time, as they rely on informal human judgement to accept or reject the null hypothesis.
</p>
<ul><li><a href="/wiki/Q%E2%80%93Q_plot" title="QQ plot">QQ plot</a>, also known as <a href="/wiki/Normal_probability_plot" title="Normal probability plot">normal probability plot</a> or <a href="/wiki/Rankit" title="Rankit">rankit</a> plot—is a plot of the sorted values from the data set against the expected values of the corresponding quantiles from the standard normal distribution. That is, it's a plot of point of the form (Φ<sup>1</sup>(<i>p<sub>k</sub></i>), <i>x</i><sub>(<i>k</i>)</sub>), where plotting points <i>p<sub>k</sub></i> are equal to <i>p<sub>k</sub></i>&#160;=&#160;(<i>k</i>&#160;&#160;<i>α</i>)/(<i>n</i>&#160;+&#160;1&#160;&#160;2<i>α</i>) and <i>α</i> is an adjustment constant, which can be anything between 0 and&#160;1. If the null hypothesis is true, the plotted points should approximately lie on a straight line.</li>
<li><a href="/wiki/P%E2%80%93P_plot" title="PP plot">PP plot</a> similar to the QQ plot, but used much less frequently. This method consists of plotting the points (Φ(<i>z</i><sub>(<i>k</i>)</sub>), <i>p<sub>k</sub></i>), where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle z_{(k)}=(x_{(k)}-{\hat {\mu }})/{\hat {\sigma }}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mstyle displaystyle="false" scriptlevel="0">
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<mi>z</mi>
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<mi>&#x03C3;<!-- σ --></mi>
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</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle \textstyle z_{(k)}=(x_{(k)}-{\hat {\mu }})/{\hat {\sigma }}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f4275afc364034dbd65279f2dd35d9a547a57ce" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:18.789ex; height:3.176ex;" alt="{\displaystyle \textstyle z_{(k)}=(x_{(k)}-{\hat {\mu }})/{\hat {\sigma }}}"></span>. For normally distributed data this plot should lie on a 45° line between (0,&#160;0) and&#160;(1,&#160;1).</li></ul>
<p><b>Goodness-of-fit tests</b>:
</p><p><i>Moment-based tests</i>:
</p>
<ul><li><a href="/wiki/D%27Agostino%27s_K-squared_test" title="D&#39;Agostino&#39;s K-squared test">D'Agostino's K-squared test</a></li>
<li><a href="/wiki/Jarque%E2%80%93Bera_test" title="JarqueBera test">JarqueBera test</a></li>
<li><a href="/wiki/Shapiro%E2%80%93Wilk_test" title="ShapiroWilk test">ShapiroWilk test</a>: This is based on the fact that the line in the QQ plot has the slope of <i>σ</i>. The test compares the least squares estimate of that slope with the value of the sample variance, and rejects the null hypothesis if these two quantities differ significantly.</li></ul>
<p><i>Tests based on the empirical distribution function</i>:
</p>
<ul><li><a href="/wiki/Anderson%E2%80%93Darling_test" title="AndersonDarling test">AndersonDarling test</a></li>
<li><a href="/wiki/Lilliefors_test" title="Lilliefors test">Lilliefors test</a> (an adaptation of the <a href="/wiki/Kolmogorov%E2%80%93Smirnov_test" title="KolmogorovSmirnov test">KolmogorovSmirnov test</a>)</li></ul>
<h3><span class="mw-headline" id="Bayesian_analysis_of_the_normal_distribution">Bayesian analysis of the normal distribution</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=39" title="Edit section: Bayesian analysis of the normal distribution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
<p>Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered:
</p>
<ul><li>Either the mean, or the variance, or neither, may be considered a fixed quantity.</li>
<li>When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the <a href="/wiki/Precision_(statistics)" title="Precision (statistics)">precision</a>, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified.</li>
<li>Both univariate and <a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">multivariate</a> cases need to be considered.</li>
<li>Either <a href="/wiki/Conjugate_prior" title="Conjugate prior">conjugate</a> or <a href="/wiki/Improper_prior" class="mw-redirect" title="Improper prior">improper</a> <a href="/wiki/Prior_distribution" class="mw-redirect" title="Prior distribution">prior distributions</a> may be placed on the unknown variables.</li>
<li>An additional set of cases occurs in <a href="/wiki/Bayesian_linear_regression" title="Bayesian linear regression">Bayesian linear regression</a>, where in the basic model the data is assumed to be normally distributed, and normal priors are placed on the <a href="/wiki/Regression_coefficient" class="mw-redirect" title="Regression coefficient">regression coefficients</a>. The resulting analysis is similar to the basic cases of <a href="/wiki/Independent_identically_distributed" class="mw-redirect" title="Independent identically distributed">independent identically distributed</a> data.</li></ul>
<p>The formulas for the non-linear-regression cases are summarized in the <a href="/wiki/Conjugate_prior" title="Conjugate prior">conjugate prior</a> article.
</p>
<h4><span class="mw-headline" id="Sum_of_two_quadratics">Sum of two quadratics</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=40" title="Edit section: Sum of two quadratics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h4>
<h5><span class="mw-headline" id="Scalar_form">Scalar form</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=41" title="Edit section: Scalar form"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h5>
<p>The following auxiliary formula is useful for simplifying the <a href="/wiki/Posterior_distribution" class="mw-redirect" title="Posterior distribution">posterior</a> update equations, which otherwise become fairly tedious.
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(x-y)^{2}+b(x-z)^{2}=(a+b)\left(x-{\frac {ay+bz}{a+b}}\right)^{2}+{\frac {ab}{a+b}}(y-z)^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
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<mfrac>
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</mrow>
<mo stretchy="false">(</mo>
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<mo>&#x2212;<!-- --></mo>
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<annotation encoding="application/x-tex">{\displaystyle a(x-y)^{2}+b(x-z)^{2}=(a+b)\left(x-{\frac {ay+bz}{a+b}}\right)^{2}+{\frac {ab}{a+b}}(y-z)^{2}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfac4114765b1f994800c9b424b82564b57ba179" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:64.839ex; height:6.509ex;" alt="a(x-y)^{2}+b(x-z)^{2}=(a+b)\left(x-{\frac {ay+bz}{a+b}}\right)^{2}+{\frac {ab}{a+b}}(y-z)^{2}"></span></dd></dl>
<p>This equation rewrites the sum of two quadratics in <i>x</i> by expanding the squares, grouping the terms in <i>x</i>, and <a href="/wiki/Completing_the_square" title="Completing the square">completing the square</a>. Note the following about the complex constant factors attached to some of the terms:
</p>
<ol><li>The factor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {ay+bz}{a+b}}}">
<semantics>
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<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
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<annotation encoding="application/x-tex">{\displaystyle {\frac {ay+bz}{a+b}}}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bd78708f583c5a7306db32820ddbd072860d318" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:8.148ex; height:5.676ex;" alt="{\frac {ay+bz}{a+b}}"></span> has the form of a <a href="/wiki/Weighted_average" class="mw-redirect" title="Weighted average">weighted average</a> of <i>y</i> and <i>z</i>.</li>
<li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {ab}{a+b}}={\frac {1}{{\frac {1}{a}}+{\frac {1}{b}}}}=(a^{-1}+b^{-1})^{-1}.}">
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<mstyle displaystyle="true" scriptlevel="0">
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</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {ab}{a+b}}={\frac {1}{{\frac {1}{a}}+{\frac {1}{b}}}}=(a^{-1}+b^{-1})^{-1}.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bd6398104a9b1bb647ee6fbc9cd7fc24843330b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:33.663ex; height:7.009ex;" alt="{\frac {ab}{a+b}}={\frac {1}{{\frac {1}{a}}+{\frac {1}{b}}}}=(a^{-1}+b^{-1})^{-1}."></span> This shows that this factor can be thought of as resulting from a situation where the <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">reciprocals</a> of quantities <i>a</i> and <i>b</i> add directly, so to combine <i>a</i> and <i>b</i> themselves, it's necessary to reciprocate, add, and reciprocate the result again to get back into the original units. This is exactly the sort of operation performed by the <a href="/wiki/Harmonic_mean" title="Harmonic mean">harmonic mean</a>, so it is not surprising that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {ab}{a+b}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>a</mi>
<mi>b</mi>
</mrow>
<mrow>
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {ab}{a+b}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bff74a1779f39a1e8f0d0dd53f7072d90924d28e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:5.904ex; height:5.676ex;" alt="{\frac {ab}{a+b}}"></span> is one-half the <a href="/wiki/Harmonic_mean" title="Harmonic mean">harmonic mean</a> of <i>a</i> and <i>b</i>.</li></ol>
<h5><span class="mw-headline" id="Vector_form">Vector form</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=42" title="Edit section: Vector form"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h5>
<p>A similar formula can be written for the sum of two vector quadratics: If <b>x</b>, <b>y</b>, <b>z</b> are vectors of length <i>k</i>, and <b>A</b> and <b>B</b> are <a href="/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric</a>, <a href="/wiki/Invertible_matrices" class="mw-redirect" title="Invertible matrices">invertible matrices</a> of size <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\times k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
<mo>&#x00D7;<!-- × --></mo>
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k\times k}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77bcf9346bcb189917b6b49c4331b4483f4a4a2c" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.263ex; height:2.176ex;" alt="k\times k"></span>, then
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&amp;(\mathbf {y} -\mathbf {x} )'\mathbf {A} (\mathbf {y} -\mathbf {x} )+(\mathbf {x} -\mathbf {z} )'\mathbf {B} (\mathbf {x} -\mathbf {z} )\\={}&amp;(\mathbf {x} -\mathbf {c} )'(\mathbf {A} +\mathbf {B} )(\mathbf {x} -\mathbf {c} )+(\mathbf {y} -\mathbf {z} )'(\mathbf {A} ^{-1}+\mathbf {B} ^{-1})^{-1}(\mathbf {y} -\mathbf {z} )\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd />
<mtd>
<mi></mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">y</mi>
</mrow>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mo>&#x2032;</mo>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">A</mi>
</mrow>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">y</mi>
</mrow>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
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<mo stretchy="false">)</mo>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
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<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">z</mi>
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<msup>
<mo stretchy="false">)</mo>
<mo>&#x2032;</mo>
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<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">B</mi>
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<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">z</mi>
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<mo stretchy="false">)</mo>
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<mtr>
<mtd>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
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<mi></mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">c</mi>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mo>&#x2032;</mo>
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<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">A</mi>
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<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">B</mi>
</mrow>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">c</mi>
</mrow>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">y</mi>
</mrow>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">z</mi>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mo>&#x2032;</mo>
</msup>
<mo stretchy="false">(</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">A</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo>+</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">B</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
</mrow>
</msup>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">y</mi>
</mrow>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">z</mi>
</mrow>
<mo stretchy="false">)</mo>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&amp;(\mathbf {y} -\mathbf {x} )'\mathbf {A} (\mathbf {y} -\mathbf {x} )+(\mathbf {x} -\mathbf {z} )'\mathbf {B} (\mathbf {x} -\mathbf {z} )\\={}&amp;(\mathbf {x} -\mathbf {c} )'(\mathbf {A} +\mathbf {B} )(\mathbf {x} -\mathbf {c} )+(\mathbf {y} -\mathbf {z} )'(\mathbf {A} ^{-1}+\mathbf {B} ^{-1})^{-1}(\mathbf {y} -\mathbf {z} )\end{aligned}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6374f98fcb11f7c1273b06c44e1c0f0b84154048" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.333ex; margin-bottom: -0.171ex; width:60.548ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}&amp;(\mathbf {y} -\mathbf {x} )&#039;\mathbf {A} (\mathbf {y} -\mathbf {x} )+(\mathbf {x} -\mathbf {z} )&#039;\mathbf {B} (\mathbf {x} -\mathbf {z} )\\={}&amp;(\mathbf {x} -\mathbf {c} )&#039;(\mathbf {A} +\mathbf {B} )(\mathbf {x} -\mathbf {c} )+(\mathbf {y} -\mathbf {z} )&#039;(\mathbf {A} ^{-1}+\mathbf {B} ^{-1})^{-1}(\mathbf {y} -\mathbf {z} )\end{aligned}}}"></span></dd></dl>
<p>where
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {c} =(\mathbf {A} +\mathbf {B} )^{-1}(\mathbf {A} \mathbf {y} +\mathbf {B} \mathbf {z} )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">c</mi>
</mrow>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">A</mi>
</mrow>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">B</mi>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">A</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">y</mi>
</mrow>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">B</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">z</mi>
</mrow>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {c} =(\mathbf {A} +\mathbf {B} )^{-1}(\mathbf {A} \mathbf {y} +\mathbf {B} \mathbf {z} )}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/267a22091cc9d9afb86fcacebcc6b842cb0e9b1b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.359ex; height:3.176ex;" alt="{\displaystyle \mathbf {c} =(\mathbf {A} +\mathbf {B} )^{-1}(\mathbf {A} \mathbf {y} +\mathbf {B} \mathbf {z} )}"></span></dd></dl>
<p>The form <b>x</b> <b>A</b> <b>x</b> is called a <a href="/wiki/Quadratic_form" title="Quadratic form">quadratic form</a> and is a <a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">scalar</a>:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} '\mathbf {A} \mathbf {x} =\sum _{i,j}a_{ij}x_{i}x_{j}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mo>&#x2032;</mo>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">A</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mo>=</mo>
<munder>
<mo>&#x2211;<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
</mrow>
</munder>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} '\mathbf {A} \mathbf {x} =\sum _{i,j}a_{ij}x_{i}x_{j}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef06ff3139875b96fe704a43bbecebacdbea460" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:19.442ex; height:5.843ex;" alt="\mathbf {x} &#039;\mathbf {A} \mathbf {x} =\sum _{i,j}a_{ij}x_{i}x_{j}"></span></dd></dl>
<p>In other words, it sums up all possible combinations of products of pairs of elements from <b>x</b>, with a separate coefficient for each. In addition, since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}x_{j}=x_{j}x_{i}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x_{i}x_{j}=x_{j}x_{i}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f2236b86202f012c661865e57b12bb725776fe1" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.836ex; height:2.343ex;" alt="x_{i}x_{j}=x_{j}x_{i}"></span>, only the sum <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{ij}+a_{ji}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>+</mo>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
<mi>i</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a_{ij}+a_{ji}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a2bf7de3e9b3347b40910df01009fb7c8989d7d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.254ex; height:2.676ex;" alt="a_{ij}+a_{ji}"></span> matters for any off-diagonal elements of <b>A</b>, and there is no loss of generality in assuming that <b>A</b> is <a href="/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric</a>. Furthermore, if <b>A</b> is symmetric, then the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} '\mathbf {A} \mathbf {y} =\mathbf {y} '\mathbf {A} \mathbf {x} .}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mo>&#x2032;</mo>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">A</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">y</mi>
</mrow>
<mo>=</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">y</mi>
</mrow>
<mo>&#x2032;</mo>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">A</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">x</mi>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} '\mathbf {A} \mathbf {y} =\mathbf {y} '\mathbf {A} \mathbf {x} .}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94db6c6c4b212d77b8363f36d5924490f424877f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.798ex; height:2.843ex;" alt="{\displaystyle \mathbf {x} &#039;\mathbf {A} \mathbf {y} =\mathbf {y} &#039;\mathbf {A} \mathbf {x} .}"></span>
</p>
<h4><span class="mw-headline" id="Sum_of_differences_from_the_mean">Sum of differences from the mean</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=43" title="Edit section: Sum of differences from the mean"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h4>
<p>Another useful formula is as follows:
<div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{n}(x_{i}-\mu )^{2}=\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}+n({\bar {x}}-\mu )^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<munderover>
<mo>&#x2211;<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munderover>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>=</mo>
<munderover>
<mo>&#x2211;<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munderover>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
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</mrow>
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<mo>&#x2212;<!-- --></mo>
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<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">&#x00AF;<!-- ¯ --></mo>
</mover>
</mrow>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>+</mo>
<mi>n</mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">&#x00AF;<!-- ¯ --></mo>
</mover>
</mrow>
</mrow>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}(x_{i}-\mu )^{2}=\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}+n({\bar {x}}-\mu )^{2}}</annotation>
</semantics>
</math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6abcabe83cd01aabf39c16b0bc67994086519d02" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:40.877ex; height:6.843ex;" alt="{\displaystyle \sum _{i=1}^{n}(x_{i}-\mu )^{2}=\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}+n({\bar {x}}-\mu )^{2}}"></div>
where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\bar {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">&#x00AF;<!-- ¯ --></mo>
</mover>
</mrow>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mi>n</mi>
</mfrac>
</mrow>
<munderover>
<mo>&#x2211;<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munderover>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\textstyle {\bar {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17005337073440fa8d7c41536f875c6cd5d1fc0e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.155ex; height:3.343ex;" alt="{\textstyle {\bar {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}.}"></span>
</p>
<h3><span class="mw-headline" id="With_known_variance">With known variance</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=44" title="Edit section: With known variance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
<p>For a set of <a href="/wiki/I.i.d." class="mw-redirect" title="I.i.d.">i.i.d.</a> normally distributed data points <b>X</b> of size <i>n</i> where each individual point <i>x</i> follows <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\sim {\mathcal {N}}(\mu ,\sigma ^{2})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo>&#x223C;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">N</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>,</mo>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x\sim {\mathcal {N}}(\mu ,\sigma ^{2})}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eec32a627a0dfefb3fcd59ce15762b8d5629a67" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.333ex; height:3.176ex;" alt="x\sim {\mathcal {N}}(\mu ,\sigma ^{2})"></span> with known <a href="/wiki/Variance" title="Variance">variance</a> σ<sup>2</sup>, the <a href="/wiki/Conjugate_prior" title="Conjugate prior">conjugate prior</a> distribution is also normally distributed.
</p><p>This can be shown more easily by rewriting the variance as the <a href="/wiki/Precision_(statistics)" title="Precision (statistics)">precision</a>, i.e. using τ = 1/σ<sup>2</sup>. Then if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\sim {\mathcal {N}}(\mu ,1/\tau )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo>&#x223C;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">N</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>,</mo>
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>&#x03C4;<!-- τ --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x\sim {\mathcal {N}}(\mu ,1/\tau )}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f5a92a6695b63641da0b546449b61e734396452" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.475ex; height:3.009ex;" alt="x\sim {\mathcal {N}}(\mu ,1/\tau )"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \sim {\mathcal {N}}(\mu _{0},1/\tau _{0}),}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>&#x03BC;<!-- μ --></mi>
<mo>&#x223C;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">N</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<msub>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>,</mo>
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<msub>
<mi>&#x03C4;<!-- τ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mu \sim {\mathcal {N}}(\mu _{0},1/\tau _{0}),}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/494cefe56f6ab7931bc91c991d89cdb788349e39" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.116ex; height:3.009ex;" alt="\mu \sim {\mathcal {N}}(\mu _{0},1/\tau _{0}),"></span> we proceed as follows.
</p><p>First, the <a href="/wiki/Likelihood_function" title="Likelihood function">likelihood function</a> is (using the formula above for the sum of differences from the mean):
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}p(\mathbf {X} \mid \mu ,\tau )&amp;=\prod _{i=1}^{n}{\sqrt {\frac {\tau }{2\pi }}}\exp \left(-{\frac {1}{2}}\tau (x_{i}-\mu )^{2}\right)\\&amp;=\left({\frac {\tau }{2\pi }}\right)^{n/2}\exp \left(-{\frac {1}{2}}\tau \sum _{i=1}^{n}(x_{i}-\mu )^{2}\right)\\&amp;=\left({\frac {\tau }{2\pi }}\right)^{n/2}\exp \left[-{\frac {1}{2}}\tau \left(\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}+n({\bar {x}}-\mu )^{2}\right)\right].\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
<mi>p</mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo>&#x2223;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>,</mo>
<mi>&#x03C4;<!-- τ --></mi>
<mo stretchy="false">)</mo>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<munderover>
<mo>&#x220F;<!-- ∏ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munderover>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mfrac>
<mi>&#x03C4;<!-- τ --></mi>
<mrow>
<mn>2</mn>
<mi>&#x03C0;<!-- π --></mi>
</mrow>
</mfrac>
</msqrt>
</mrow>
<mi>exp</mi>
<mo>&#x2061;<!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mi>&#x03C4;<!-- τ --></mi>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>)</mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd />
<mtd>
<mi></mi>
<mo>=</mo>
<msup>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>&#x03C4;<!-- τ --></mi>
<mrow>
<mn>2</mn>
<mi>&#x03C0;<!-- π --></mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
</mrow>
</msup>
<mi>exp</mi>
<mo>&#x2061;<!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mi>&#x03C4;<!-- τ --></mi>
<munderover>
<mo>&#x2211;<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munderover>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>)</mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd />
<mtd>
<mi></mi>
<mo>=</mo>
<msup>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>&#x03C4;<!-- τ --></mi>
<mrow>
<mn>2</mn>
<mi>&#x03C0;<!-- π --></mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
</mrow>
</msup>
<mi>exp</mi>
<mo>&#x2061;<!-- --></mo>
<mrow>
<mo>[</mo>
<mrow>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mi>&#x03C4;<!-- τ --></mi>
<mrow>
<mo>(</mo>
<mrow>
<munderover>
<mo>&#x2211;<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munderover>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">&#x00AF;<!-- ¯ --></mo>
</mover>
</mrow>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>+</mo>
<mi>n</mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">&#x00AF;<!-- ¯ --></mo>
</mover>
</mrow>
</mrow>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
<mo>.</mo>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}p(\mathbf {X} \mid \mu ,\tau )&amp;=\prod _{i=1}^{n}{\sqrt {\frac {\tau }{2\pi }}}\exp \left(-{\frac {1}{2}}\tau (x_{i}-\mu )^{2}\right)\\&amp;=\left({\frac {\tau }{2\pi }}\right)^{n/2}\exp \left(-{\frac {1}{2}}\tau \sum _{i=1}^{n}(x_{i}-\mu )^{2}\right)\\&amp;=\left({\frac {\tau }{2\pi }}\right)^{n/2}\exp \left[-{\frac {1}{2}}\tau \left(\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}+n({\bar {x}}-\mu )^{2}\right)\right].\end{aligned}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2bcd1c34520a24e29b758a0f7427e79e9d8a414" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -10.505ex; width:64.954ex; height:22.176ex;" alt="{\displaystyle {\begin{aligned}p(\mathbf {X} \mid \mu ,\tau )&amp;=\prod _{i=1}^{n}{\sqrt {\frac {\tau }{2\pi }}}\exp \left(-{\frac {1}{2}}\tau (x_{i}-\mu )^{2}\right)\\&amp;=\left({\frac {\tau }{2\pi }}\right)^{n/2}\exp \left(-{\frac {1}{2}}\tau \sum _{i=1}^{n}(x_{i}-\mu )^{2}\right)\\&amp;=\left({\frac {\tau }{2\pi }}\right)^{n/2}\exp \left[-{\frac {1}{2}}\tau \left(\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}+n({\bar {x}}-\mu )^{2}\right)\right].\end{aligned}}}"></span></dd></dl>
<p>Then, we proceed as follows:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}p(\mu \mid \mathbf {X} )&amp;\propto p(\mathbf {X} \mid \mu )p(\mu )\\&amp;=\left({\frac {\tau }{2\pi }}\right)^{n/2}\exp \left[-{\frac {1}{2}}\tau \left(\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}+n({\bar {x}}-\mu )^{2}\right)\right]{\sqrt {\frac {\tau _{0}}{2\pi }}}\exp \left(-{\frac {1}{2}}\tau _{0}(\mu -\mu _{0})^{2}\right)\\&amp;\propto \exp \left(-{\frac {1}{2}}\left(\tau \left(\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}+n({\bar {x}}-\mu )^{2}\right)+\tau _{0}(\mu -\mu _{0})^{2}\right)\right)\\&amp;\propto \exp \left(-{\frac {1}{2}}\left(n\tau ({\bar {x}}-\mu )^{2}+\tau _{0}(\mu -\mu _{0})^{2}\right)\right)\\&amp;=\exp \left(-{\frac {1}{2}}(n\tau +\tau _{0})\left(\mu -{\dfrac {n\tau {\bar {x}}+\tau _{0}\mu _{0}}{n\tau +\tau _{0}}}\right)^{2}+{\frac {n\tau \tau _{0}}{n\tau +\tau _{0}}}({\bar {x}}-\mu _{0})^{2}\right)\\&amp;\propto \exp \left(-{\frac {1}{2}}(n\tau +\tau _{0})\left(\mu -{\dfrac {n\tau {\bar {x}}+\tau _{0}\mu _{0}}{n\tau +\tau _{0}}}\right)^{2}\right)\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
<mi>p</mi>
<mo stretchy="false">(</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>&#x2223;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo stretchy="false">)</mo>
</mtd>
<mtd>
<mi></mi>
<mo>&#x221D;<!-- ∝ --></mo>
<mi>p</mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo>&#x2223;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo stretchy="false">)</mo>
<mi>p</mi>
<mo stretchy="false">(</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo stretchy="false">)</mo>
</mtd>
</mtr>
<mtr>
<mtd />
<mtd>
<mi></mi>
<mo>=</mo>
<msup>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>&#x03C4;<!-- τ --></mi>
<mrow>
<mn>2</mn>
<mi>&#x03C0;<!-- π --></mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
</mrow>
</msup>
<mi>exp</mi>
<mo>&#x2061;<!-- --></mo>
<mrow>
<mo>[</mo>
<mrow>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mi>&#x03C4;<!-- τ --></mi>
<mrow>
<mo>(</mo>
<mrow>
<munderover>
<mo>&#x2211;<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munderover>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">&#x00AF;<!-- ¯ --></mo>
</mover>
</mrow>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>+</mo>
<mi>n</mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">&#x00AF;<!-- ¯ --></mo>
</mover>
</mrow>
</mrow>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mfrac>
<msub>
<mi>&#x03C4;<!-- τ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mrow>
<mn>2</mn>
<mi>&#x03C0;<!-- π --></mi>
</mrow>
</mfrac>
</msqrt>
</mrow>
<mi>exp</mi>
<mo>&#x2061;<!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<msub>
<mi>&#x03C4;<!-- τ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>&#x2212;<!-- --></mo>
<msub>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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</msub>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msup>
</mrow>
<mo>)</mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd />
<mtd>
<mi></mi>
<mo>&#x221D;<!-- ∝ --></mo>
<mi>exp</mi>
<mo>&#x2061;<!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>&#x03C4;<!-- τ --></mi>
<mrow>
<mo>(</mo>
<mrow>
<munderover>
<mo>&#x2211;<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
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<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
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</munderover>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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</msub>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">&#x00AF;<!-- ¯ --></mo>
</mover>
</mrow>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>+</mo>
<mi>n</mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">&#x00AF;<!-- ¯ --></mo>
</mover>
</mrow>
</mrow>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<msub>
<mi>&#x03C4;<!-- τ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>&#x2212;<!-- --></mo>
<msub>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd />
<mtd>
<mi></mi>
<mo>&#x221D;<!-- ∝ --></mo>
<mi>exp</mi>
<mo>&#x2061;<!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>n</mi>
<mi>&#x03C4;<!-- τ --></mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">&#x00AF;<!-- ¯ --></mo>
</mover>
</mrow>
</mrow>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>+</mo>
<msub>
<mi>&#x03C4;<!-- τ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>&#x2212;<!-- --></mo>
<msub>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd />
<mtd>
<mi></mi>
<mo>=</mo>
<mi>exp</mi>
<mo>&#x2061;<!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mi>&#x03C4;<!-- τ --></mi>
<mo>+</mo>
<msub>
<mi>&#x03C4;<!-- τ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>&#x03BC;<!-- μ --></mi>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mfrac>
<mrow>
<mi>n</mi>
<mi>&#x03C4;<!-- τ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">&#x00AF;<!-- ¯ --></mo>
</mover>
</mrow>
</mrow>
<mo>+</mo>
<msub>
<mi>&#x03C4;<!-- τ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<msub>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mrow>
<mrow>
<mi>n</mi>
<mi>&#x03C4;<!-- τ --></mi>
<mo>+</mo>
<msub>
<mi>&#x03C4;<!-- τ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mrow>
</mfrac>
</mstyle>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>n</mi>
<mi>&#x03C4;<!-- τ --></mi>
<msub>
<mi>&#x03C4;<!-- τ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mrow>
<mrow>
<mi>n</mi>
<mi>&#x03C4;<!-- τ --></mi>
<mo>+</mo>
<msub>
<mi>&#x03C4;<!-- τ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">&#x00AF;<!-- ¯ --></mo>
</mover>
</mrow>
</mrow>
<mo>&#x2212;<!-- --></mo>
<msub>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>)</mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd />
<mtd>
<mi></mi>
<mo>&#x221D;<!-- ∝ --></mo>
<mi>exp</mi>
<mo>&#x2061;<!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mi>&#x03C4;<!-- τ --></mi>
<mo>+</mo>
<msub>
<mi>&#x03C4;<!-- τ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>&#x03BC;<!-- μ --></mi>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mfrac>
<mrow>
<mi>n</mi>
<mi>&#x03C4;<!-- τ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">&#x00AF;<!-- ¯ --></mo>
</mover>
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</mrow>
<mo>+</mo>
<msub>
<mi>&#x03C4;<!-- τ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<msub>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mrow>
<mrow>
<mi>n</mi>
<mi>&#x03C4;<!-- τ --></mi>
<mo>+</mo>
<msub>
<mi>&#x03C4;<!-- τ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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<mo>)</mo>
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<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}p(\mu \mid \mathbf {X} )&amp;\propto p(\mathbf {X} \mid \mu )p(\mu )\\&amp;=\left({\frac {\tau }{2\pi }}\right)^{n/2}\exp \left[-{\frac {1}{2}}\tau \left(\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}+n({\bar {x}}-\mu )^{2}\right)\right]{\sqrt {\frac {\tau _{0}}{2\pi }}}\exp \left(-{\frac {1}{2}}\tau _{0}(\mu -\mu _{0})^{2}\right)\\&amp;\propto \exp \left(-{\frac {1}{2}}\left(\tau \left(\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}+n({\bar {x}}-\mu )^{2}\right)+\tau _{0}(\mu -\mu _{0})^{2}\right)\right)\\&amp;\propto \exp \left(-{\frac {1}{2}}\left(n\tau ({\bar {x}}-\mu )^{2}+\tau _{0}(\mu -\mu _{0})^{2}\right)\right)\\&amp;=\exp \left(-{\frac {1}{2}}(n\tau +\tau _{0})\left(\mu -{\dfrac {n\tau {\bar {x}}+\tau _{0}\mu _{0}}{n\tau +\tau _{0}}}\right)^{2}+{\frac {n\tau \tau _{0}}{n\tau +\tau _{0}}}({\bar {x}}-\mu _{0})^{2}\right)\\&amp;\propto \exp \left(-{\frac {1}{2}}(n\tau +\tau _{0})\left(\mu -{\dfrac {n\tau {\bar {x}}+\tau _{0}\mu _{0}}{n\tau +\tau _{0}}}\right)^{2}\right)\end{aligned}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96e309ead00fbc8603eced5342aa5df534522d6a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -19.338ex; width:90.525ex; height:39.843ex;" alt="{\displaystyle {\begin{aligned}p(\mu \mid \mathbf {X} )&amp;\propto p(\mathbf {X} \mid \mu )p(\mu )\\&amp;=\left({\frac {\tau }{2\pi }}\right)^{n/2}\exp \left[-{\frac {1}{2}}\tau \left(\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}+n({\bar {x}}-\mu )^{2}\right)\right]{\sqrt {\frac {\tau _{0}}{2\pi }}}\exp \left(-{\frac {1}{2}}\tau _{0}(\mu -\mu _{0})^{2}\right)\\&amp;\propto \exp \left(-{\frac {1}{2}}\left(\tau \left(\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}+n({\bar {x}}-\mu )^{2}\right)+\tau _{0}(\mu -\mu _{0})^{2}\right)\right)\\&amp;\propto \exp \left(-{\frac {1}{2}}\left(n\tau ({\bar {x}}-\mu )^{2}+\tau _{0}(\mu -\mu _{0})^{2}\right)\right)\\&amp;=\exp \left(-{\frac {1}{2}}(n\tau +\tau _{0})\left(\mu -{\dfrac {n\tau {\bar {x}}+\tau _{0}\mu _{0}}{n\tau +\tau _{0}}}\right)^{2}+{\frac {n\tau \tau _{0}}{n\tau +\tau _{0}}}({\bar {x}}-\mu _{0})^{2}\right)\\&amp;\propto \exp \left(-{\frac {1}{2}}(n\tau +\tau _{0})\left(\mu -{\dfrac {n\tau {\bar {x}}+\tau _{0}\mu _{0}}{n\tau +\tau _{0}}}\right)^{2}\right)\end{aligned}}}"></span></dd></dl>
<p>In the above derivation, we used the formula above for the sum of two quadratics and eliminated all constant factors not involving&#160;<i>μ</i>. The result is the <a href="/wiki/Kernel_(statistics)" title="Kernel (statistics)">kernel</a> of a normal distribution, with mean <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {n\tau {\bar {x}}+\tau _{0}\mu _{0}}{n\tau +\tau _{0}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>n</mi>
<mi>&#x03C4;<!-- τ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">&#x00AF;<!-- ¯ --></mo>
</mover>
</mrow>
</mrow>
<mo>+</mo>
<msub>
<mi>&#x03C4;<!-- τ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<msub>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mrow>
<mrow>
<mi>n</mi>
<mi>&#x03C4;<!-- τ --></mi>
<mo>+</mo>
<msub>
<mi>&#x03C4;<!-- τ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {n\tau {\bar {x}}+\tau _{0}\mu _{0}}{n\tau +\tau _{0}}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdacf96584fa3c673a6efc97a94da6cc92ee6a03" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:12.129ex; height:5.509ex;" alt="{\frac {n\tau {\bar {x}}+\tau _{0}\mu _{0}}{n\tau +\tau _{0}}}"></span> and precision <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\tau +\tau _{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mi>&#x03C4;<!-- τ --></mi>
<mo>+</mo>
<msub>
<mi>&#x03C4;<!-- τ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n\tau +\tau _{0}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f597302f3954facee30e057129dddc94fe898667" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.507ex; height:2.343ex;" alt="n\tau +\tau _{0}"></span>, i.e.
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(\mu \mid \mathbf {X} )\sim {\mathcal {N}}\left({\frac {n\tau {\bar {x}}+\tau _{0}\mu _{0}}{n\tau +\tau _{0}}},{\frac {1}{n\tau +\tau _{0}}}\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
<mo stretchy="false">(</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>&#x2223;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo stretchy="false">)</mo>
<mo>&#x223C;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">N</mi>
</mrow>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>n</mi>
<mi>&#x03C4;<!-- τ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">&#x00AF;<!-- ¯ --></mo>
</mover>
</mrow>
</mrow>
<mo>+</mo>
<msub>
<mi>&#x03C4;<!-- τ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<msub>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mrow>
<mrow>
<mi>n</mi>
<mi>&#x03C4;<!-- τ --></mi>
<mo>+</mo>
<msub>
<mi>&#x03C4;<!-- τ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mi>n</mi>
<mi>&#x03C4;<!-- τ --></mi>
<mo>+</mo>
<msub>
<mi>&#x03C4;<!-- τ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mrow>
</mfrac>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p(\mu \mid \mathbf {X} )\sim {\mathcal {N}}\left({\frac {n\tau {\bar {x}}+\tau _{0}\mu _{0}}{n\tau +\tau _{0}}},{\frac {1}{n\tau +\tau _{0}}}\right)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a45b361f59d044be9a7d87bf92514795f38419c8" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; margin-left: -0.089ex; width:39.115ex; height:6.176ex;" alt="p(\mu \mid \mathbf {X} )\sim {\mathcal {N}}\left({\frac {n\tau {\bar {x}}+\tau _{0}\mu _{0}}{n\tau +\tau _{0}}},{\frac {1}{n\tau +\tau _{0}}}\right)"></span></dd></dl>
<p>This can be written as a set of Bayesian update equations for the posterior parameters in terms of the prior parameters:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\tau _{0}'&amp;=\tau _{0}+n\tau \\[5pt]\mu _{0}'&amp;={\frac {n\tau {\bar {x}}+\tau _{0}\mu _{0}}{n\tau +\tau _{0}}}\\[5pt]{\bar {x}}&amp;={\frac {1}{n}}\sum _{i=1}^{n}x_{i}\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="0.8em 0.8em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
<msubsup>
<mi>&#x03C4;<!-- τ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mo>&#x2032;</mo>
</msubsup>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<msub>
<mi>&#x03C4;<!-- τ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>+</mo>
<mi>n</mi>
<mi>&#x03C4;<!-- τ --></mi>
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<mtr>
<mtd>
<msubsup>
<mi>&#x03BC;<!-- μ --></mi>
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<mi>&#x03C4;<!-- τ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\tau _{0}'&amp;=\tau _{0}+n\tau \\[5pt]\mu _{0}'&amp;={\frac {n\tau {\bar {x}}+\tau _{0}\mu _{0}}{n\tau +\tau _{0}}}\\[5pt]{\bar {x}}&amp;={\frac {1}{n}}\sum _{i=1}^{n}x_{i}\end{aligned}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6cfbdf504b1a9ce4cbe79561b4ae983fdf7271d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -8.505ex; width:18.435ex; height:18.176ex;" alt="{\displaystyle {\begin{aligned}\tau _{0}&#039;&amp;=\tau _{0}+n\tau \\[5pt]\mu _{0}&#039;&amp;={\frac {n\tau {\bar {x}}+\tau _{0}\mu _{0}}{n\tau +\tau _{0}}}\\[5pt]{\bar {x}}&amp;={\frac {1}{n}}\sum _{i=1}^{n}x_{i}\end{aligned}}}"></span></dd></dl>
<p>That is, to combine <i>n</i> data points with total precision of <i>nτ</i> (or equivalently, total variance of <i>n</i>/<i>σ</i><sup>2</sup>) and mean of values <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {x}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">&#x00AF;<!-- ¯ --></mo>
</mover>
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</mrow>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\bar {x}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/466e03e1c9533b4dab1b9949dad393883f385d80" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.009ex;" alt="{\bar {x}}"></span>, derive a new total precision simply by adding the total precision of the data to the prior total precision, and form a new mean through a <i>precision-weighted average</i>, i.e. a <a href="/wiki/Weighted_average" class="mw-redirect" title="Weighted average">weighted average</a> of the data mean and the prior mean, each weighted by the associated total precision. This makes logical sense if the precision is thought of as indicating the certainty of the observations: In the distribution of the posterior mean, each of the input components is weighted by its certainty, and the certainty of this distribution is the sum of the individual certainties. (For the intuition of this, compare the expression "the whole is (or is not) greater than the sum of its parts". In addition, consider that the knowledge of the posterior comes from a combination of the knowledge of the prior and likelihood, so it makes sense that we are more certain of it than of either of its components.)
</p><p>The above formula reveals why it is more convenient to do <a href="/wiki/Bayesian_analysis" class="mw-redirect" title="Bayesian analysis">Bayesian analysis</a> of <a href="/wiki/Conjugate_prior" title="Conjugate prior">conjugate priors</a> for the normal distribution in terms of the precision. The posterior precision is simply the sum of the prior and likelihood precisions, and the posterior mean is computed through a precision-weighted average, as described above. The same formulas can be written in terms of variance by reciprocating all the precisions, yielding the more ugly formulas
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\sigma _{0}^{2}}'&amp;={\frac {1}{{\frac {n}{\sigma ^{2}}}+{\frac {1}{\sigma _{0}^{2}}}}}\\[5pt]\mu _{0}'&amp;={\frac {{\frac {n{\bar {x}}}{\sigma ^{2}}}+{\frac {\mu _{0}}{\sigma _{0}^{2}}}}{{\frac {n}{\sigma ^{2}}}+{\frac {1}{\sigma _{0}^{2}}}}}\\[5pt]{\bar {x}}&amp;={\frac {1}{n}}\sum _{i=1}^{n}x_{i}\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="0.8em 0.8em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
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<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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<mo>&#x2032;</mo>
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<mtd>
<mi></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
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<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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<mo>+</mo>
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<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
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<mrow class="MJX-TeXAtom-ORD">
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<mtd>
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<mi>&#x03BC;<!-- μ --></mi>
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<mrow class="MJX-TeXAtom-ORD">
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<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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<mrow class="MJX-TeXAtom-ORD">
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<mrow>
<mrow class="MJX-TeXAtom-ORD">
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<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
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<mo>+</mo>
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<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\sigma _{0}^{2}}'&amp;={\frac {1}{{\frac {n}{\sigma ^{2}}}+{\frac {1}{\sigma _{0}^{2}}}}}\\[5pt]\mu _{0}'&amp;={\frac {{\frac {n{\bar {x}}}{\sigma ^{2}}}+{\frac {\mu _{0}}{\sigma _{0}^{2}}}}{{\frac {n}{\sigma ^{2}}}+{\frac {1}{\sigma _{0}^{2}}}}}\\[5pt]{\bar {x}}&amp;={\frac {1}{n}}\sum _{i=1}^{n}x_{i}\end{aligned}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea454c8840683777ce8192d9ae63068c63962858" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -13.171ex; width:16.023ex; height:27.509ex;" alt="{\displaystyle {\begin{aligned}{\sigma _{0}^{2}}&#039;&amp;={\frac {1}{{\frac {n}{\sigma ^{2}}}+{\frac {1}{\sigma _{0}^{2}}}}}\\[5pt]\mu _{0}&#039;&amp;={\frac {{\frac {n{\bar {x}}}{\sigma ^{2}}}+{\frac {\mu _{0}}{\sigma _{0}^{2}}}}{{\frac {n}{\sigma ^{2}}}+{\frac {1}{\sigma _{0}^{2}}}}}\\[5pt]{\bar {x}}&amp;={\frac {1}{n}}\sum _{i=1}^{n}x_{i}\end{aligned}}}"></span></dd></dl>
<h4><span class="mw-headline" id="With_known_mean">With known mean</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=45" title="Edit section: With known mean"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h4>
<p>For a set of <a href="/wiki/I.i.d." class="mw-redirect" title="I.i.d.">i.i.d.</a> normally distributed data points <b>X</b> of size <i>n</i> where each individual point <i>x</i> follows <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\sim {\mathcal {N}}(\mu ,\sigma ^{2})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo>&#x223C;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">N</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>,</mo>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x\sim {\mathcal {N}}(\mu ,\sigma ^{2})}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eec32a627a0dfefb3fcd59ce15762b8d5629a67" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.333ex; height:3.176ex;" alt="x\sim {\mathcal {N}}(\mu ,\sigma ^{2})"></span> with known mean μ, the <a href="/wiki/Conjugate_prior" title="Conjugate prior">conjugate prior</a> of the <a href="/wiki/Variance" title="Variance">variance</a> has an <a href="/wiki/Inverse_gamma_distribution" class="mw-redirect" title="Inverse gamma distribution">inverse gamma distribution</a> or a <a href="/wiki/Scaled_inverse_chi-squared_distribution" title="Scaled inverse chi-squared distribution">scaled inverse chi-squared distribution</a>. The two are equivalent except for having different <a href="/wiki/Parameter" title="Parameter">parameterizations</a>. Although the inverse gamma is more commonly used, we use the scaled inverse chi-squared for the sake of convenience. The prior for σ<sup>2</sup> is as follows:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(\sigma ^{2}\mid \nu _{0},\sigma _{0}^{2})={\frac {(\sigma _{0}^{2}{\frac {\nu _{0}}{2}})^{\nu _{0}/2}}{\Gamma \left({\frac {\nu _{0}}{2}}\right)}}~{\frac {\exp \left[{\frac {-\nu _{0}\sigma _{0}^{2}}{2\sigma ^{2}}}\right]}{(\sigma ^{2})^{1+{\frac {\nu _{0}}{2}}}}}\propto {\frac {\exp \left[{\frac {-\nu _{0}\sigma _{0}^{2}}{2\sigma ^{2}}}\right]}{(\sigma ^{2})^{1+{\frac {\nu _{0}}{2}}}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
<mo stretchy="false">(</mo>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
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<mn>2</mn>
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</msup>
<mo>&#x2223;<!-- --></mo>
<msub>
<mi>&#x03BD;<!-- ν --></mi>
<mrow class="MJX-TeXAtom-ORD">
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<mo>,</mo>
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<mrow class="MJX-TeXAtom-ORD">
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<mrow class="MJX-TeXAtom-ORD">
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<msubsup>
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<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msub>
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<mrow class="MJX-TeXAtom-ORD">
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</msub>
<mn>2</mn>
</mfrac>
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<msup>
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<msub>
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</msub>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
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<mn>2</mn>
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<mi mathvariant="normal">&#x0393;<!-- Γ --></mi>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msub>
<mi>&#x03BD;<!-- ν --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
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</mrow>
</mfrac>
</mrow>
<mtext>&#xA0;</mtext>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>exp</mi>
<mo>&#x2061;<!-- --></mo>
<mrow>
<mo>[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo>&#x2212;<!-- --></mo>
<msub>
<mi>&#x03BD;<!-- ν --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
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</mrow>
<mrow>
<mn>2</mn>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msup>
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</mrow>
<mo>]</mo>
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<mrow>
<mo stretchy="false">(</mo>
<msup>
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<mrow class="MJX-TeXAtom-ORD">
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</msup>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msub>
<mi>&#x03BD;<!-- ν --></mi>
<mrow class="MJX-TeXAtom-ORD">
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</msub>
<mn>2</mn>
</mfrac>
</mrow>
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</msup>
</mrow>
</mfrac>
</mrow>
<mo>&#x221D;<!-- ∝ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>exp</mi>
<mo>&#x2061;<!-- --></mo>
<mrow>
<mo>[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo>&#x2212;<!-- --></mo>
<msub>
<mi>&#x03BD;<!-- ν --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
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<mrow>
<mn>2</mn>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mo>]</mo>
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</mrow>
<mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mo>+</mo>
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<mfrac>
<msub>
<mi>&#x03BD;<!-- ν --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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</msub>
<mn>2</mn>
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</msup>
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</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p(\sigma ^{2}\mid \nu _{0},\sigma _{0}^{2})={\frac {(\sigma _{0}^{2}{\frac {\nu _{0}}{2}})^{\nu _{0}/2}}{\Gamma \left({\frac {\nu _{0}}{2}}\right)}}~{\frac {\exp \left[{\frac {-\nu _{0}\sigma _{0}^{2}}{2\sigma ^{2}}}\right]}{(\sigma ^{2})^{1+{\frac {\nu _{0}}{2}}}}}\propto {\frac {\exp \left[{\frac {-\nu _{0}\sigma _{0}^{2}}{2\sigma ^{2}}}\right]}{(\sigma ^{2})^{1+{\frac {\nu _{0}}{2}}}}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef2528fe4774a93087d4adae570ef9ab84707f52" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -4.005ex; margin-left: -0.089ex; width:55.568ex; height:11.009ex;" alt="{\displaystyle p(\sigma ^{2}\mid \nu _{0},\sigma _{0}^{2})={\frac {(\sigma _{0}^{2}{\frac {\nu _{0}}{2}})^{\nu _{0}/2}}{\Gamma \left({\frac {\nu _{0}}{2}}\right)}}~{\frac {\exp \left[{\frac {-\nu _{0}\sigma _{0}^{2}}{2\sigma ^{2}}}\right]}{(\sigma ^{2})^{1+{\frac {\nu _{0}}{2}}}}}\propto {\frac {\exp \left[{\frac {-\nu _{0}\sigma _{0}^{2}}{2\sigma ^{2}}}\right]}{(\sigma ^{2})^{1+{\frac {\nu _{0}}{2}}}}}}"></span></dd></dl>
<p>The <a href="/wiki/Likelihood_function" title="Likelihood function">likelihood function</a> from above, written in terms of the variance, is:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}p(\mathbf {X} \mid \mu ,\sigma ^{2})&amp;=\left({\frac {1}{2\pi \sigma ^{2}}}\right)^{n/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\sum _{i=1}^{n}(x_{i}-\mu )^{2}\right]\\&amp;=\left({\frac {1}{2\pi \sigma ^{2}}}\right)^{n/2}\exp \left[-{\frac {S}{2\sigma ^{2}}}\right]\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
<mi>p</mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo>&#x2223;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>,</mo>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
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<mtd>
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<mrow>
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<mi>&#x03C0;<!-- π --></mi>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
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</mrow>
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</mrow>
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<mn>2</mn>
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</mrow>
<mo>)</mo>
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<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
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</mtd>
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</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}p(\mathbf {X} \mid \mu ,\sigma ^{2})&amp;=\left({\frac {1}{2\pi \sigma ^{2}}}\right)^{n/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\sum _{i=1}^{n}(x_{i}-\mu )^{2}\right]\\&amp;=\left({\frac {1}{2\pi \sigma ^{2}}}\right)^{n/2}\exp \left[-{\frac {S}{2\sigma ^{2}}}\right]\end{aligned}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc06aa31588bba03e4748f8f345f0638a75dc156" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:53.423ex; height:14.509ex;" alt="{\displaystyle {\begin{aligned}p(\mathbf {X} \mid \mu ,\sigma ^{2})&amp;=\left({\frac {1}{2\pi \sigma ^{2}}}\right)^{n/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\sum _{i=1}^{n}(x_{i}-\mu )^{2}\right]\\&amp;=\left({\frac {1}{2\pi \sigma ^{2}}}\right)^{n/2}\exp \left[-{\frac {S}{2\sigma ^{2}}}\right]\end{aligned}}}"></span></dd></dl>
<p>where
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\sum _{i=1}^{n}(x_{i}-\mu )^{2}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>S</mi>
<mo>=</mo>
<munderover>
<mo>&#x2211;<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
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<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
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<msup>
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<mrow class="MJX-TeXAtom-ORD">
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</msup>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle S=\sum _{i=1}^{n}(x_{i}-\mu )^{2}.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56adf28a77173ce852c7de7eeee102b2f6895b39" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.834ex; height:6.843ex;" alt="S=\sum _{i=1}^{n}(x_{i}-\mu )^{2}."></span></dd></dl>
<p>Then:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}p(\sigma ^{2}\mid \mathbf {X} )&amp;\propto p(\mathbf {X} \mid \sigma ^{2})p(\sigma ^{2})\\&amp;=\left({\frac {1}{2\pi \sigma ^{2}}}\right)^{n/2}\exp \left[-{\frac {S}{2\sigma ^{2}}}\right]{\frac {(\sigma _{0}^{2}{\frac {\nu _{0}}{2}})^{\frac {\nu _{0}}{2}}}{\Gamma \left({\frac {\nu _{0}}{2}}\right)}}~{\frac {\exp \left[{\frac {-\nu _{0}\sigma _{0}^{2}}{2\sigma ^{2}}}\right]}{(\sigma ^{2})^{1+{\frac {\nu _{0}}{2}}}}}\\&amp;\propto \left({\frac {1}{\sigma ^{2}}}\right)^{n/2}{\frac {1}{(\sigma ^{2})^{1+{\frac {\nu _{0}}{2}}}}}\exp \left[-{\frac {S}{2\sigma ^{2}}}+{\frac {-\nu _{0}\sigma _{0}^{2}}{2\sigma ^{2}}}\right]\\&amp;={\frac {1}{(\sigma ^{2})^{1+{\frac {\nu _{0}+n}{2}}}}}\exp \left[-{\frac {\nu _{0}\sigma _{0}^{2}+S}{2\sigma ^{2}}}\right]\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
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</mrow>
<mi>exp</mi>
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<mrow>
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<mn>2</mn>
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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}p(\sigma ^{2}\mid \mathbf {X} )&amp;\propto p(\mathbf {X} \mid \sigma ^{2})p(\sigma ^{2})\\&amp;=\left({\frac {1}{2\pi \sigma ^{2}}}\right)^{n/2}\exp \left[-{\frac {S}{2\sigma ^{2}}}\right]{\frac {(\sigma _{0}^{2}{\frac {\nu _{0}}{2}})^{\frac {\nu _{0}}{2}}}{\Gamma \left({\frac {\nu _{0}}{2}}\right)}}~{\frac {\exp \left[{\frac {-\nu _{0}\sigma _{0}^{2}}{2\sigma ^{2}}}\right]}{(\sigma ^{2})^{1+{\frac {\nu _{0}}{2}}}}}\\&amp;\propto \left({\frac {1}{\sigma ^{2}}}\right)^{n/2}{\frac {1}{(\sigma ^{2})^{1+{\frac {\nu _{0}}{2}}}}}\exp \left[-{\frac {S}{2\sigma ^{2}}}+{\frac {-\nu _{0}\sigma _{0}^{2}}{2\sigma ^{2}}}\right]\\&amp;={\frac {1}{(\sigma ^{2})^{1+{\frac {\nu _{0}+n}{2}}}}}\exp \left[-{\frac {\nu _{0}\sigma _{0}^{2}+S}{2\sigma ^{2}}}\right]\end{aligned}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/381c1b93f6dc76e2cdca9f3f1f77132dd51dc55f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -15.171ex; width:60.748ex; height:31.509ex;" alt="{\displaystyle {\begin{aligned}p(\sigma ^{2}\mid \mathbf {X} )&amp;\propto p(\mathbf {X} \mid \sigma ^{2})p(\sigma ^{2})\\&amp;=\left({\frac {1}{2\pi \sigma ^{2}}}\right)^{n/2}\exp \left[-{\frac {S}{2\sigma ^{2}}}\right]{\frac {(\sigma _{0}^{2}{\frac {\nu _{0}}{2}})^{\frac {\nu _{0}}{2}}}{\Gamma \left({\frac {\nu _{0}}{2}}\right)}}~{\frac {\exp \left[{\frac {-\nu _{0}\sigma _{0}^{2}}{2\sigma ^{2}}}\right]}{(\sigma ^{2})^{1+{\frac {\nu _{0}}{2}}}}}\\&amp;\propto \left({\frac {1}{\sigma ^{2}}}\right)^{n/2}{\frac {1}{(\sigma ^{2})^{1+{\frac {\nu _{0}}{2}}}}}\exp \left[-{\frac {S}{2\sigma ^{2}}}+{\frac {-\nu _{0}\sigma _{0}^{2}}{2\sigma ^{2}}}\right]\\&amp;={\frac {1}{(\sigma ^{2})^{1+{\frac {\nu _{0}+n}{2}}}}}\exp \left[-{\frac {\nu _{0}\sigma _{0}^{2}+S}{2\sigma ^{2}}}\right]\end{aligned}}}"></span></dd></dl>
<p>The above is also a scaled inverse chi-squared distribution where
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\nu _{0}'&amp;=\nu _{0}+n\\\nu _{0}'{\sigma _{0}^{2}}'&amp;=\nu _{0}\sigma _{0}^{2}+\sum _{i=1}^{n}(x_{i}-\mu )^{2}\end{aligned}}}">
<semantics>
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<mtd>
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<mi>&#x03BD;<!-- ν --></mi>
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<msup>
<mrow class="MJX-TeXAtom-ORD">
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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<mrow class="MJX-TeXAtom-ORD">
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<mo>&#x2211;<!-- ∑ --></mo>
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<mrow class="MJX-TeXAtom-ORD">
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<mo stretchy="false">(</mo>
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<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\nu _{0}'&amp;=\nu _{0}+n\\\nu _{0}'{\sigma _{0}^{2}}'&amp;=\nu _{0}\sigma _{0}^{2}+\sum _{i=1}^{n}(x_{i}-\mu )^{2}\end{aligned}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1d9cea4f20a8750894be82fb32d617284c433fd" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:29.14ex; height:10.176ex;" alt="{\begin{aligned}\nu _{0}&#039;&amp;=\nu _{0}+n\\\nu _{0}&#039;{\sigma _{0}^{2}}&#039;&amp;=\nu _{0}\sigma _{0}^{2}+\sum _{i=1}^{n}(x_{i}-\mu )^{2}\end{aligned}}"></span></dd></dl>
<p>or equivalently
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\nu _{0}'&amp;=\nu _{0}+n\\{\sigma _{0}^{2}}'&amp;={\frac {\nu _{0}\sigma _{0}^{2}+\sum _{i=1}^{n}(x_{i}-\mu )^{2}}{\nu _{0}+n}}\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
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<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
<msubsup>
<mi>&#x03BD;<!-- ν --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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<mo>&#x2032;</mo>
</msubsup>
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<mtd>
<mi></mi>
<mo>=</mo>
<msub>
<mi>&#x03BD;<!-- ν --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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<mo>+</mo>
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<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msubsup>
</mrow>
<mo>&#x2032;</mo>
</msup>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msub>
<mi>&#x03BD;<!-- ν --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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</msub>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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<mo>+</mo>
<munderover>
<mo>&#x2211;<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munderover>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mrow>
<msub>
<mi>&#x03BD;<!-- ν --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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</msub>
<mo>+</mo>
<mi>n</mi>
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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\nu _{0}'&amp;=\nu _{0}+n\\{\sigma _{0}^{2}}'&amp;={\frac {\nu _{0}\sigma _{0}^{2}+\sum _{i=1}^{n}(x_{i}-\mu )^{2}}{\nu _{0}+n}}\end{aligned}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/192be53c5d9d249b2ef7ca5622430b689f1aee64" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:29.772ex; height:9.843ex;" alt="{\begin{aligned}\nu _{0}&#039;&amp;=\nu _{0}+n\\{\sigma _{0}^{2}}&#039;&amp;={\frac {\nu _{0}\sigma _{0}^{2}+\sum _{i=1}^{n}(x_{i}-\mu )^{2}}{\nu _{0}+n}}\end{aligned}}"></span></dd></dl>
<p>Reparameterizing in terms of an <a href="/wiki/Inverse_gamma_distribution" class="mw-redirect" title="Inverse gamma distribution">inverse gamma distribution</a>, the result is:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\alpha '&amp;=\alpha +{\frac {n}{2}}\\\beta '&amp;=\beta +{\frac {\sum _{i=1}^{n}(x_{i}-\mu )^{2}}{2}}\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
<msup>
<mi>&#x03B1;<!-- α --></mi>
<mo>&#x2032;</mo>
</msup>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mi>&#x03B1;<!-- α --></mi>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>n</mi>
<mn>2</mn>
</mfrac>
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<mtr>
<mtd>
<msup>
<mi>&#x03B2;<!-- β --></mi>
<mo>&#x2032;</mo>
</msup>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mi>&#x03B2;<!-- β --></mi>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<munderover>
<mo>&#x2211;<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munderover>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mn>2</mn>
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</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\alpha '&amp;=\alpha +{\frac {n}{2}}\\\beta '&amp;=\beta +{\frac {\sum _{i=1}^{n}(x_{i}-\mu )^{2}}{2}}\end{aligned}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6242673d0e1932e640fa7ebb2167edbb20535f35" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:25.62ex; height:10.843ex;" alt="{\begin{aligned}\alpha &#039;&amp;=\alpha +{\frac {n}{2}}\\\beta &#039;&amp;=\beta +{\frac {\sum _{i=1}^{n}(x_{i}-\mu )^{2}}{2}}\end{aligned}}"></span></dd></dl>
<h4><span class="mw-headline" id="With_unknown_mean_and_unknown_variance">With unknown mean and unknown variance</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=46" title="Edit section: With unknown mean and unknown variance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h4>
<p>For a set of <a href="/wiki/I.i.d." class="mw-redirect" title="I.i.d.">i.i.d.</a> normally distributed data points <b>X</b> of size <i>n</i> where each individual point <i>x</i> follows <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\sim {\mathcal {N}}(\mu ,\sigma ^{2})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo>&#x223C;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">N</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>,</mo>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x\sim {\mathcal {N}}(\mu ,\sigma ^{2})}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eec32a627a0dfefb3fcd59ce15762b8d5629a67" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.333ex; height:3.176ex;" alt="x\sim {\mathcal {N}}(\mu ,\sigma ^{2})"></span> with unknown mean μ and unknown <a href="/wiki/Variance" title="Variance">variance</a> σ<sup>2</sup>, a combined (multivariate) <a href="/wiki/Conjugate_prior" title="Conjugate prior">conjugate prior</a> is placed over the mean and variance, consisting of a <a href="/wiki/Normal-inverse-gamma_distribution" title="Normal-inverse-gamma distribution">normal-inverse-gamma distribution</a>.
Logically, this originates as follows:
</p>
<ol><li>From the analysis of the case with unknown mean but known variance, we see that the update equations involve <a href="/wiki/Sufficient_statistic" title="Sufficient statistic">sufficient statistics</a> computed from the data consisting of the mean of the data points and the total variance of the data points, computed in turn from the known variance divided by the number of data points.</li>
<li>From the analysis of the case with unknown variance but known mean, we see that the update equations involve sufficient statistics over the data consisting of the number of data points and <a href="/wiki/Sum_of_squared_deviations" class="mw-redirect" title="Sum of squared deviations">sum of squared deviations</a>.</li>
<li>Keep in mind that the posterior update values serve as the prior distribution when further data is handled. Thus, we should logically think of our priors in terms of the sufficient statistics just described, with the same semantics kept in mind as much as possible.</li>
<li>To handle the case where both mean and variance are unknown, we could place independent priors over the mean and variance, with fixed estimates of the average mean, total variance, number of data points used to compute the variance prior, and sum of squared deviations. Note however that in reality, the total variance of the mean depends on the unknown variance, and the sum of squared deviations that goes into the variance prior (appears to) depend on the unknown mean. In practice, the latter dependence is relatively unimportant: Shifting the actual mean shifts the generated points by an equal amount, and on average the squared deviations will remain the same. This is not the case, however, with the total variance of the mean: As the unknown variance increases, the total variance of the mean will increase proportionately, and we would like to capture this dependence.</li>
<li>This suggests that we create a <i>conditional prior</i> of the mean on the unknown variance, with a hyperparameter specifying the mean of the <a href="/wiki/Pseudo-observation" class="mw-redirect" title="Pseudo-observation">pseudo-observations</a> associated with the prior, and another parameter specifying the number of pseudo-observations. This number serves as a scaling parameter on the variance, making it possible to control the overall variance of the mean relative to the actual variance parameter. The prior for the variance also has two hyperparameters, one specifying the sum of squared deviations of the pseudo-observations associated with the prior, and another specifying once again the number of pseudo-observations. Each of the priors has a hyperparameter specifying the number of pseudo-observations, and in each case this controls the relative variance of that prior. These are given as two separate hyperparameters so that the variance (aka the confidence) of the two priors can be controlled separately.</li>
<li>This leads immediately to the <a href="/wiki/Normal-inverse-gamma_distribution" title="Normal-inverse-gamma distribution">normal-inverse-gamma distribution</a>, which is the product of the two distributions just defined, with <a href="/wiki/Conjugate_prior" title="Conjugate prior">conjugate priors</a> used (an <a href="/wiki/Inverse_gamma_distribution" class="mw-redirect" title="Inverse gamma distribution">inverse gamma distribution</a> over the variance, and a normal distribution over the mean, <i>conditional</i> on the variance) and with the same four parameters just defined.</li></ol>
<p>The priors are normally defined as follows:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}p(\mu \mid \sigma ^{2};\mu _{0},n_{0})&amp;\sim {\mathcal {N}}(\mu _{0},\sigma ^{2}/n_{0})\\p(\sigma ^{2};\nu _{0},\sigma _{0}^{2})&amp;\sim I\chi ^{2}(\nu _{0},\sigma _{0}^{2})=IG(\nu _{0}/2,\nu _{0}\sigma _{0}^{2}/2)\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
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<mi>p</mi>
<mo stretchy="false">(</mo>
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<mtd>
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<mo>,</mo>
<msub>
<mi>&#x03BD;<!-- ν --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}p(\mu \mid \sigma ^{2};\mu _{0},n_{0})&amp;\sim {\mathcal {N}}(\mu _{0},\sigma ^{2}/n_{0})\\p(\sigma ^{2};\nu _{0},\sigma _{0}^{2})&amp;\sim I\chi ^{2}(\nu _{0},\sigma _{0}^{2})=IG(\nu _{0}/2,\nu _{0}\sigma _{0}^{2}/2)\end{aligned}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bab8dee515d3208f73dd85d1cb46706e3a9097f9" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:51.017ex; height:6.509ex;" alt="{\begin{aligned}p(\mu \mid \sigma ^{2};\mu _{0},n_{0})&amp;\sim {\mathcal {N}}(\mu _{0},\sigma ^{2}/n_{0})\\p(\sigma ^{2};\nu _{0},\sigma _{0}^{2})&amp;\sim I\chi ^{2}(\nu _{0},\sigma _{0}^{2})=IG(\nu _{0}/2,\nu _{0}\sigma _{0}^{2}/2)\end{aligned}}"></span></dd></dl>
<p>The update equations can be derived, and look as follows:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\bar {x}}&amp;={\frac {1}{n}}\sum _{i=1}^{n}x_{i}\\\mu _{0}'&amp;={\frac {n_{0}\mu _{0}+n{\bar {x}}}{n_{0}+n}}\\n_{0}'&amp;=n_{0}+n\\\nu _{0}'&amp;=\nu _{0}+n\\\nu _{0}'{\sigma _{0}^{2}}'&amp;=\nu _{0}\sigma _{0}^{2}+\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}+{\frac {n_{0}n}{n_{0}+n}}(\mu _{0}-{\bar {x}})^{2}\end{aligned}}}">
<semantics>
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<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">&#x00AF;<!-- ¯ --></mo>
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<mtd>
<mi></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mi>n</mi>
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<mo>&#x2211;<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munderover>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
</mtd>
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<mtr>
<mtd>
<msubsup>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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<mo>&#x2032;</mo>
</msubsup>
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<mi></mi>
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<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
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<msub>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>+</mo>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">&#x00AF;<!-- ¯ --></mo>
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</mrow>
<mrow>
<msub>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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<mo>+</mo>
<mi>n</mi>
</mrow>
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</mrow>
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</mtr>
<mtr>
<mtd>
<msubsup>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mo>&#x2032;</mo>
</msubsup>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<msub>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>+</mo>
<mi>n</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<msubsup>
<mi>&#x03BD;<!-- ν --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mo>&#x2032;</mo>
</msubsup>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<msub>
<mi>&#x03BD;<!-- ν --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>+</mo>
<mi>n</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<msubsup>
<mi>&#x03BD;<!-- ν --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mo>&#x2032;</mo>
</msubsup>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
</mrow>
<mo>&#x2032;</mo>
</msup>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<msub>
<mi>&#x03BD;<!-- ν --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo>+</mo>
<munderover>
<mo>&#x2211;<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munderover>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">&#x00AF;<!-- ¯ --></mo>
</mover>
</mrow>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msub>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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</msub>
<mi>n</mi>
</mrow>
<mrow>
<msub>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>+</mo>
<mi>n</mi>
</mrow>
</mfrac>
</mrow>
<mo stretchy="false">(</mo>
<msub>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>x</mi>
<mo stretchy="false">&#x00AF;<!-- ¯ --></mo>
</mover>
</mrow>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\bar {x}}&amp;={\frac {1}{n}}\sum _{i=1}^{n}x_{i}\\\mu _{0}'&amp;={\frac {n_{0}\mu _{0}+n{\bar {x}}}{n_{0}+n}}\\n_{0}'&amp;=n_{0}+n\\\nu _{0}'&amp;=\nu _{0}+n\\\nu _{0}'{\sigma _{0}^{2}}'&amp;=\nu _{0}\sigma _{0}^{2}+\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}+{\frac {n_{0}n}{n_{0}+n}}(\mu _{0}-{\bar {x}})^{2}\end{aligned}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/673b045d8322e2ce9e1ecc33c00585873b85547a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -12.671ex; width:48.918ex; height:26.509ex;" alt="{\displaystyle {\begin{aligned}{\bar {x}}&amp;={\frac {1}{n}}\sum _{i=1}^{n}x_{i}\\\mu _{0}&#039;&amp;={\frac {n_{0}\mu _{0}+n{\bar {x}}}{n_{0}+n}}\\n_{0}&#039;&amp;=n_{0}+n\\\nu _{0}&#039;&amp;=\nu _{0}+n\\\nu _{0}&#039;{\sigma _{0}^{2}}&#039;&amp;=\nu _{0}\sigma _{0}^{2}+\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}+{\frac {n_{0}n}{n_{0}+n}}(\mu _{0}-{\bar {x}})^{2}\end{aligned}}}"></span></dd></dl>
<p>The respective numbers of pseudo-observations add the number of actual observations to them. The new mean hyperparameter is once again a weighted average, this time weighted by the relative numbers of observations. Finally, the update for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu _{0}'{\sigma _{0}^{2}}'}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mi>&#x03BD;<!-- ν --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mo>&#x2032;</mo>
</msubsup>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
</mrow>
<mo>&#x2032;</mo>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \nu _{0}'{\sigma _{0}^{2}}'}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70e03cda663483a2b055eebe89eee1fbc65e4456" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.272ex; height:3.509ex;" alt="\nu _{0}&#039;{\sigma _{0}^{2}}&#039;"></span> is similar to the case with known mean, but in this case the sum of squared deviations is taken with respect to the observed data mean rather than the true mean, and as a result a new interaction term needs to be added to take care of the additional error source stemming from the deviation between prior and data mean.
</p>
<style data-mw-deduplicate="TemplateStyles:r1174254338">.mw-parser-output .math_proof{border:thin solid #aaa;margin:1em 2em;padding:0.5em 1em 0.4em}@media(max-width:500px){.mw-parser-output .math_proof{margin:1em 0;padding:0.5em 0.5em 0.4em}}</style><div class="math_proof" style=""><strong>Proof</strong>
<p>The prior distributions are
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}p(\mu \mid \sigma ^{2};\mu _{0},n_{0})&amp;\sim {\mathcal {N}}(\mu _{0},\sigma ^{2}/n_{0})={\frac {1}{\sqrt {2\pi {\frac {\sigma ^{2}}{n_{0}}}}}}\exp \left(-{\frac {n_{0}}{2\sigma ^{2}}}(\mu -\mu _{0})^{2}\right)\\&amp;\propto (\sigma ^{2})^{-1/2}\exp \left(-{\frac {n_{0}}{2\sigma ^{2}}}(\mu -\mu _{0})^{2}\right)\\p(\sigma ^{2};\nu _{0},\sigma _{0}^{2})&amp;\sim I\chi ^{2}(\nu _{0},\sigma _{0}^{2})=IG(\nu _{0}/2,\nu _{0}\sigma _{0}^{2}/2)\\&amp;={\frac {(\sigma _{0}^{2}\nu _{0}/2)^{\nu _{0}/2}}{\Gamma (\nu _{0}/2)}}~{\frac {\exp \left[{\frac {-\nu _{0}\sigma _{0}^{2}}{2\sigma ^{2}}}\right]}{(\sigma ^{2})^{1+\nu _{0}/2}}}\\&amp;\propto {(\sigma ^{2})^{-(1+\nu _{0}/2)}}\exp \left[{\frac {-\nu _{0}\sigma _{0}^{2}}{2\sigma ^{2}}}\right].\end{aligned}}}">
<semantics>
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<mtd>
<mi>p</mi>
<mo stretchy="false">(</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>&#x2223;<!-- --></mo>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>;</mo>
<msub>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mtd>
<mtd>
<mi></mi>
<mo>&#x223C;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">N</mi>
</mrow>
</mrow>
<mo stretchy="false">(</mo>
<msub>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>,</mo>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<msub>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<msqrt>
<mn>2</mn>
<mi>&#x03C0;<!-- π --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<msub>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mfrac>
</mrow>
</msqrt>
</mfrac>
</mrow>
<mi>exp</mi>
<mo>&#x2061;<!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msub>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mrow>
<mn>2</mn>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mo stretchy="false">(</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>&#x2212;<!-- --></mo>
<msub>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>)</mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd />
<mtd>
<mi></mi>
<mo>&#x221D;<!-- ∝ --></mo>
<mo stretchy="false">(</mo>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
</mrow>
</msup>
<mi>exp</mi>
<mo>&#x2061;<!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msub>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mrow>
<mn>2</mn>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mo stretchy="false">(</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>&#x2212;<!-- --></mo>
<msub>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>)</mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>p</mi>
<mo stretchy="false">(</mo>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>;</mo>
<msub>
<mi>&#x03BD;<!-- ν --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>,</mo>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo stretchy="false">)</mo>
</mtd>
<mtd>
<mi></mi>
<mo>&#x223C;<!-- --></mo>
<mi>I</mi>
<msup>
<mi>&#x03C7;<!-- χ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<msub>
<mi>&#x03BD;<!-- ν --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>,</mo>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>I</mi>
<mi>G</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>&#x03BD;<!-- ν --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
<mo>,</mo>
<msub>
<mi>&#x03BD;<!-- ν --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mtd>
</mtr>
<mtr>
<mtd />
<mtd>
<mi></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<msub>
<mi>&#x03BD;<!-- ν --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<msub>
<mi>&#x03BD;<!-- ν --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mrow>
<mi mathvariant="normal">&#x0393;<!-- Γ --></mi>
<mo stretchy="false">(</mo>
<msub>
<mi>&#x03BD;<!-- ν --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
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<mn>2</mn>
<mo stretchy="false">)</mo>
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</mfrac>
</mrow>
<mtext>&#xA0;</mtext>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>exp</mi>
<mo>&#x2061;<!-- --></mo>
<mrow>
<mo>[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo>&#x2212;<!-- --></mo>
<msub>
<mi>&#x03BD;<!-- ν --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
</mrow>
<mrow>
<mn>2</mn>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mo>]</mo>
</mrow>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mo>+</mo>
<msub>
<mi>&#x03BD;<!-- ν --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
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</msup>
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</mtd>
</mtr>
<mtr>
<mtd />
<mtd>
<mi></mi>
<mo>&#x221D;<!-- ∝ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<mo stretchy="false">(</mo>
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<msub>
<mi>&#x03BD;<!-- ν --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
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</msub>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
<mi>exp</mi>
<mo>&#x2061;<!-- --></mo>
<mrow>
<mo>[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo>&#x2212;<!-- --></mo>
<msub>
<mi>&#x03BD;<!-- ν --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
</mrow>
<mrow>
<mn>2</mn>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mo>]</mo>
</mrow>
<mo>.</mo>
</mtd>
</mtr>
</mtable>
</mrow>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}p(\mu \mid \sigma ^{2};\mu _{0},n_{0})&amp;\sim {\mathcal {N}}(\mu _{0},\sigma ^{2}/n_{0})={\frac {1}{\sqrt {2\pi {\frac {\sigma ^{2}}{n_{0}}}}}}\exp \left(-{\frac {n_{0}}{2\sigma ^{2}}}(\mu -\mu _{0})^{2}\right)\\&amp;\propto (\sigma ^{2})^{-1/2}\exp \left(-{\frac {n_{0}}{2\sigma ^{2}}}(\mu -\mu _{0})^{2}\right)\\p(\sigma ^{2};\nu _{0},\sigma _{0}^{2})&amp;\sim I\chi ^{2}(\nu _{0},\sigma _{0}^{2})=IG(\nu _{0}/2,\nu _{0}\sigma _{0}^{2}/2)\\&amp;={\frac {(\sigma _{0}^{2}\nu _{0}/2)^{\nu _{0}/2}}{\Gamma (\nu _{0}/2)}}~{\frac {\exp \left[{\frac {-\nu _{0}\sigma _{0}^{2}}{2\sigma ^{2}}}\right]}{(\sigma ^{2})^{1+\nu _{0}/2}}}\\&amp;\propto {(\sigma ^{2})^{-(1+\nu _{0}/2)}}\exp \left[{\frac {-\nu _{0}\sigma _{0}^{2}}{2\sigma ^{2}}}\right].\end{aligned}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf7afb8e3b63fb1526171840344b32458e55cf8b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -17.338ex; width:67.616ex; height:35.843ex;" alt="{\displaystyle {\begin{aligned}p(\mu \mid \sigma ^{2};\mu _{0},n_{0})&amp;\sim {\mathcal {N}}(\mu _{0},\sigma ^{2}/n_{0})={\frac {1}{\sqrt {2\pi {\frac {\sigma ^{2}}{n_{0}}}}}}\exp \left(-{\frac {n_{0}}{2\sigma ^{2}}}(\mu -\mu _{0})^{2}\right)\\&amp;\propto (\sigma ^{2})^{-1/2}\exp \left(-{\frac {n_{0}}{2\sigma ^{2}}}(\mu -\mu _{0})^{2}\right)\\p(\sigma ^{2};\nu _{0},\sigma _{0}^{2})&amp;\sim I\chi ^{2}(\nu _{0},\sigma _{0}^{2})=IG(\nu _{0}/2,\nu _{0}\sigma _{0}^{2}/2)\\&amp;={\frac {(\sigma _{0}^{2}\nu _{0}/2)^{\nu _{0}/2}}{\Gamma (\nu _{0}/2)}}~{\frac {\exp \left[{\frac {-\nu _{0}\sigma _{0}^{2}}{2\sigma ^{2}}}\right]}{(\sigma ^{2})^{1+\nu _{0}/2}}}\\&amp;\propto {(\sigma ^{2})^{-(1+\nu _{0}/2)}}\exp \left[{\frac {-\nu _{0}\sigma _{0}^{2}}{2\sigma ^{2}}}\right].\end{aligned}}}"></span></dd></dl>
<p>Therefore, the joint prior is
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}p(\mu ,\sigma ^{2};\mu _{0},n_{0},\nu _{0},\sigma _{0}^{2})&amp;=p(\mu \mid \sigma ^{2};\mu _{0},n_{0})\,p(\sigma ^{2};\nu _{0},\sigma _{0}^{2})\\&amp;\propto (\sigma ^{2})^{-(\nu _{0}+3)/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\nu _{0}\sigma _{0}^{2}+n_{0}(\mu -\mu _{0})^{2}\right)\right].\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
<mi>p</mi>
<mo stretchy="false">(</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>,</mo>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>;</mo>
<msub>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>&#x03BD;<!-- ν --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>,</mo>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo stretchy="false">)</mo>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mi>p</mi>
<mo stretchy="false">(</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>&#x2223;<!-- --></mo>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>;</mo>
<msub>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace" />
<mi>p</mi>
<mo stretchy="false">(</mo>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>;</mo>
<msub>
<mi>&#x03BD;<!-- ν --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>,</mo>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msubsup>
<mo stretchy="false">)</mo>
</mtd>
</mtr>
<mtr>
<mtd />
<mtd>
<mi></mi>
<mo>&#x221D;<!-- ∝ --></mo>
<mo stretchy="false">(</mo>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>&#x2212;<!-- --></mo>
<mo stretchy="false">(</mo>
<msub>
<mi>&#x03BD;<!-- ν --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>+</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
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</msup>
<mi>exp</mi>
<mo>&#x2061;<!-- --></mo>
<mrow>
<mo>[</mo>
<mrow>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<msub>
<mi>&#x03BD;<!-- ν --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<msubsup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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<mo>+</mo>
<msub>
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>&#x2212;<!-- --></mo>
<msub>
<mi>&#x03BC;<!-- μ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>)</mo>
</mrow>
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</mrow>
<mo>.</mo>
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</mtr>
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</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}p(\mu ,\sigma ^{2};\mu _{0},n_{0},\nu _{0},\sigma _{0}^{2})&amp;=p(\mu \mid \sigma ^{2};\mu _{0},n_{0})\,p(\sigma ^{2};\nu _{0},\sigma _{0}^{2})\\&amp;\propto (\sigma ^{2})^{-(\nu _{0}+3)/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\nu _{0}\sigma _{0}^{2}+n_{0}(\mu -\mu _{0})^{2}\right)\right].\end{aligned}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6f808161077baef3854dbfd90b870698d721090" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:72.838ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}p(\mu ,\sigma ^{2};\mu _{0},n_{0},\nu _{0},\sigma _{0}^{2})&amp;=p(\mu \mid \sigma ^{2};\mu _{0},n_{0})\,p(\sigma ^{2};\nu _{0},\sigma _{0}^{2})\\&amp;\propto (\sigma ^{2})^{-(\nu _{0}+3)/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\nu _{0}\sigma _{0}^{2}+n_{0}(\mu -\mu _{0})^{2}\right)\right].\end{aligned}}}"></span></dd></dl>
<p>The <a href="/wiki/Likelihood_function" title="Likelihood function">likelihood function</a> from the section above with known variance is:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}p(\mathbf {X} \mid \mu ,\sigma ^{2})&amp;=\left({\frac {1}{2\pi \sigma ^{2}}}\right)^{n/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\sum _{i=1}^{n}(x_{i}-\mu )^{2}\right)\right]\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
<mi>p</mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo>&#x2223;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>,</mo>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<msup>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<mi>&#x03C0;<!-- π --></mi>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
</mrow>
</msup>
<mi>exp</mi>
<mo>&#x2061;<!-- --></mo>
<mrow>
<mo>[</mo>
<mrow>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<munderover>
<mo>&#x2211;<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</munderover>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo>&#x2212;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}p(\mathbf {X} \mid \mu ,\sigma ^{2})&amp;=\left({\frac {1}{2\pi \sigma ^{2}}}\right)^{n/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\sum _{i=1}^{n}(x_{i}-\mu )^{2}\right)\right]\end{aligned}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3d77342aadcb34c5d84418cecaefdb52842b6b7" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:57.104ex; height:7.509ex;" alt="{\begin{aligned}p(\mathbf {X} \mid \mu ,\sigma ^{2})&amp;=\left({\frac {1}{2\pi \sigma ^{2}}}\right)^{n/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\sum _{i=1}^{n}(x_{i}-\mu )^{2}\right)\right]\end{aligned}}"></span></dd></dl>
<p>Writing it in terms of variance rather than precision, we get:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}p(\mathbf {X} \mid \mu ,\sigma ^{2})&amp;=\left({\frac {1}{2\pi \sigma ^{2}}}\right)^{n/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}+n({\bar {x}}-\mu )^{2}\right)\right]\\&amp;\propto {\sigma ^{2}}^{-n/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(S+n({\bar {x}}-\mu )^{2}\right)\right]\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
<mi>p</mi>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">X</mi>
</mrow>
<mo>&#x2223;<!-- --></mo>
<mi>&#x03BC;<!-- μ --></mi>
<mo>,</mo>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mtd>
<mtd>
<mi></mi>
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<msup>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<mi>&#x03C0;<!-- π --></mi>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
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</mfrac>
</mrow>
<mo>)</mo>
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<mi>n</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
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<mn>2</mn>
</mrow>
</msup>
<mi>exp</mi>
<mo>&#x2061;<!-- --></mo>
<mrow>
<mo>[</mo>
<mrow>
<mo>&#x2212;<!-- --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<msup>
<mi>&#x03C3;<!-- σ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<munderover>
<mo>&#x2211;<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
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<mrow class="MJX-TeXAtom-ORD">
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<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}p(\mathbf {X} \mid \mu ,\sigma ^{2})&amp;=\left({\frac {1}{2\pi \sigma ^{2}}}\right)^{n/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}+n({\bar {x}}-\mu )^{2}\right)\right]\\&amp;\propto {\sigma ^{2}}^{-n/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(S+n({\bar {x}}-\mu )^{2}\right)\right]\end{aligned}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29b915f070b522a1e9f419be05624c86c854ca14" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:69.703ex; height:13.843ex;" alt="{\begin{aligned}p(\mathbf {X} \mid \mu ,\sigma ^{2})&amp;=\left({\frac {1}{2\pi \sigma ^{2}}}\right)^{n/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}+n({\bar {x}}-\mu )^{2}\right)\right]\\&amp;\propto {\sigma ^{2}}^{-n/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(S+n({\bar {x}}-\mu )^{2}\right)\right]\end{aligned}}"></span></dd></dl>
<p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle S=\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}.}">
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<annotation encoding="application/x-tex">{\textstyle S=\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}.}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02a32ce18746fffb539a3316846c9a6b40695c31" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.762ex; height:3.176ex;" alt="{\textstyle S=\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}.}"></span>
</p><p>Therefore, the posterior is (dropping the hyperparameters as conditioning factors):
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}p(\mu ,\sigma ^{2}\mid \mathbf {X} )&amp;\propto p(\mu ,\sigma ^{2})\,p(\mathbf {X} \mid \mu ,\sigma ^{2})\\&amp;\propto (\sigma ^{2})^{-(\nu _{0}+3)/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\nu _{0}\sigma _{0}^{2}+n_{0}(\mu -\mu _{0})^{2}\right)\right]{\sigma ^{2}}^{-n/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(S+n({\bar {x}}-\mu )^{2}\right)\right]\\&amp;=(\sigma ^{2})^{-(\nu _{0}+n+3)/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\nu _{0}\sigma _{0}^{2}+S+n_{0}(\mu -\mu _{0})^{2}+n({\bar {x}}-\mu )^{2}\right)\right]\\&amp;=(\sigma ^{2})^{-(\nu _{0}+n+3)/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\nu _{0}\sigma _{0}^{2}+S+{\frac {n_{0}n}{n_{0}+n}}(\mu _{0}-{\bar {x}})^{2}+(n_{0}+n)\left(\mu -{\frac {n_{0}\mu _{0}+n{\bar {x}}}{n_{0}+n}}\right)^{2}\right)\right]\\&amp;\propto (\sigma ^{2})^{-1/2}\exp \left[-{\frac {n_{0}+n}{2\sigma ^{2}}}\left(\mu -{\frac {n_{0}\mu _{0}+n{\bar {x}}}{n_{0}+n}}\right)^{2}\right]\\&amp;\quad \times (\sigma ^{2})^{-(\nu _{0}/2+n/2+1)}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\nu _{0}\sigma _{0}^{2}+S+{\frac {n_{0}n}{n_{0}+n}}(\mu _{0}-{\bar {x}})^{2}\right)\right]\\&amp;={\mathcal {N}}_{\mu \mid \sigma ^{2}}\left({\frac {n_{0}\mu _{0}+n{\bar {x}}}{n_{0}+n}},{\frac {\sigma ^{2}}{n_{0}+n}}\right)\cdot {\rm {IG}}_{\sigma ^{2}}\left({\frac {1}{2}}(\nu _{0}+n),{\frac {1}{2}}\left(\nu _{0}\sigma _{0}^{2}+S+{\frac {n_{0}n}{n_{0}+n}}(\mu _{0}-{\bar {x}})^{2}\right)\right).\end{aligned}}}">
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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}p(\mu ,\sigma ^{2}\mid \mathbf {X} )&amp;\propto p(\mu ,\sigma ^{2})\,p(\mathbf {X} \mid \mu ,\sigma ^{2})\\&amp;\propto (\sigma ^{2})^{-(\nu _{0}+3)/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\nu _{0}\sigma _{0}^{2}+n_{0}(\mu -\mu _{0})^{2}\right)\right]{\sigma ^{2}}^{-n/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(S+n({\bar {x}}-\mu )^{2}\right)\right]\\&amp;=(\sigma ^{2})^{-(\nu _{0}+n+3)/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\nu _{0}\sigma _{0}^{2}+S+n_{0}(\mu -\mu _{0})^{2}+n({\bar {x}}-\mu )^{2}\right)\right]\\&amp;=(\sigma ^{2})^{-(\nu _{0}+n+3)/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\nu _{0}\sigma _{0}^{2}+S+{\frac {n_{0}n}{n_{0}+n}}(\mu _{0}-{\bar {x}})^{2}+(n_{0}+n)\left(\mu -{\frac {n_{0}\mu _{0}+n{\bar {x}}}{n_{0}+n}}\right)^{2}\right)\right]\\&amp;\propto (\sigma ^{2})^{-1/2}\exp \left[-{\frac {n_{0}+n}{2\sigma ^{2}}}\left(\mu -{\frac {n_{0}\mu _{0}+n{\bar {x}}}{n_{0}+n}}\right)^{2}\right]\\&amp;\quad \times (\sigma ^{2})^{-(\nu _{0}/2+n/2+1)}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\nu _{0}\sigma _{0}^{2}+S+{\frac {n_{0}n}{n_{0}+n}}(\mu _{0}-{\bar {x}})^{2}\right)\right]\\&amp;={\mathcal {N}}_{\mu \mid \sigma ^{2}}\left({\frac {n_{0}\mu _{0}+n{\bar {x}}}{n_{0}+n}},{\frac {\sigma ^{2}}{n_{0}+n}}\right)\cdot {\rm {IG}}_{\sigma ^{2}}\left({\frac {1}{2}}(\nu _{0}+n),{\frac {1}{2}}\left(\nu _{0}\sigma _{0}^{2}+S+{\frac {n_{0}n}{n_{0}+n}}(\mu _{0}-{\bar {x}})^{2}\right)\right).\end{aligned}}}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cad9489034d77d53c12c7ee6044f712cfdb77831" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -21.338ex; width:107.287ex; height:43.843ex;" alt="{\begin{aligned}p(\mu ,\sigma ^{2}\mid \mathbf {X} )&amp;\propto p(\mu ,\sigma ^{2})\,p(\mathbf {X} \mid \mu ,\sigma ^{2})\\&amp;\propto (\sigma ^{2})^{-(\nu _{0}+3)/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\nu _{0}\sigma _{0}^{2}+n_{0}(\mu -\mu _{0})^{2}\right)\right]{\sigma ^{2}}^{-n/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(S+n({\bar {x}}-\mu )^{2}\right)\right]\\&amp;=(\sigma ^{2})^{-(\nu _{0}+n+3)/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\nu _{0}\sigma _{0}^{2}+S+n_{0}(\mu -\mu _{0})^{2}+n({\bar {x}}-\mu )^{2}\right)\right]\\&amp;=(\sigma ^{2})^{-(\nu _{0}+n+3)/2}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\nu _{0}\sigma _{0}^{2}+S+{\frac {n_{0}n}{n_{0}+n}}(\mu _{0}-{\bar {x}})^{2}+(n_{0}+n)\left(\mu -{\frac {n_{0}\mu _{0}+n{\bar {x}}}{n_{0}+n}}\right)^{2}\right)\right]\\&amp;\propto (\sigma ^{2})^{-1/2}\exp \left[-{\frac {n_{0}+n}{2\sigma ^{2}}}\left(\mu -{\frac {n_{0}\mu _{0}+n{\bar {x}}}{n_{0}+n}}\right)^{2}\right]\\&amp;\quad \times (\sigma ^{2})^{-(\nu _{0}/2+n/2+1)}\exp \left[-{\frac {1}{2\sigma ^{2}}}\left(\nu _{0}\sigma _{0}^{2}+S+{\frac {n_{0}n}{n_{0}+n}}(\mu _{0}-{\bar {x}})^{2}\right)\right]\\&amp;={\mathcal {N}}_{\mu \mid \sigma ^{2}}\left({\frac {n_{0}\mu _{0}+n{\bar {x}}}{n_{0}+n}},{\frac {\sigma ^{2}}{n_{0}+n}}\right)\cdot {\rm {IG}}_{\sigma ^{2}}\left({\frac {1}{2}}(\nu _{0}+n),{\frac {1}{2}}\left(\nu _{0}\sigma _{0}^{2}+S+{\frac {n_{0}n}{n_{0}+n}}(\mu _{0}-{\bar {x}})^{2}\right)\right).\end{aligned}}"></span></dd></dl>
<p>In other words, the posterior distribution has the form of a product of a normal distribution over <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle p(\mu |\sigma ^{2})}">
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9734bce3c2a0076e7414aadcb56bed5e7c09ec9a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:7.501ex; height:3.009ex;" alt="{\textstyle p(\mu |\sigma ^{2})}"></span> times an inverse gamma distribution over <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle p(\sigma ^{2})}">
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23d06552135aaa8a49d59cd28e4fc282a247bdd6" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:5.453ex; height:3.009ex;" alt="{\textstyle p(\sigma ^{2})}"></span>, with parameters that are the same as the update equations above.
</p>
</div>
<p><br />
</p>
<h2><span class="mw-headline" id="Occurrence_and_applications">Occurrence and applications</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=47" title="Edit section: Occurrence and applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h2>
<p>The occurrence of normal distribution in practical problems can be loosely classified into four categories:
</p>
<ol><li>Exactly normal distributions;</li>
<li>Approximately normal laws, for example when such approximation is justified by the <a href="/wiki/Central_limit_theorem" title="Central limit theorem">central limit theorem</a>; and</li>
<li>Distributions modeled as normal the normal distribution being the distribution with <a href="/wiki/Principle_of_maximum_entropy" title="Principle of maximum entropy">maximum entropy</a> for a given mean and variance.</li>
<li>Regression problems the normal distribution being found after systematic effects have been modeled sufficiently well.</li></ol>
<h3><span class="mw-headline" id="Exact_normality">Exact normality</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=48" title="Edit section: Exact normality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
<figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:QHarmonicOscillator.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/b/bb/QHarmonicOscillator.png" decoding="async" width="164" height="139" class="mw-file-element" data-file-width="164" data-file-height="139" /></a><figcaption>The ground state of a <a href="/wiki/Quantum_harmonic_oscillator" title="Quantum harmonic oscillator">quantum harmonic oscillator</a> has the <a href="/wiki/Gaussian_distribution" class="mw-redirect" title="Gaussian distribution">Gaussian distribution</a>.</figcaption></figure>
<p>Certain quantities in <a href="/wiki/Physics" title="Physics">physics</a> are distributed normally, as was first demonstrated by <a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">James Clerk Maxwell</a>. Examples of such quantities are:
</p>
<ul><li>Probability density function of a ground state in a <a href="/wiki/Quantum_harmonic_oscillator" title="Quantum harmonic oscillator">quantum harmonic oscillator</a>.</li>
<li>The position of a particle that experiences <a href="/wiki/Diffusion" title="Diffusion">diffusion</a>. If initially the particle is located at a specific point (that is its probability distribution is the <a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a>), then after time <i>t</i> its location is described by a normal distribution with variance <i>t</i>, which satisfies the <a href="/wiki/Diffusion_equation" title="Diffusion equation">diffusion equation</a>&#160;<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial }{\partial t}}f(x,t)={\frac {1}{2}}{\frac {\partial ^{2}}{\partial x^{2}}}f(x,t)}">
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/290df4d132d669a2940e351bebcbe802e0d9202c" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:25.211ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial }{\partial t}}f(x,t)={\frac {1}{2}}{\frac {\partial ^{2}}{\partial x^{2}}}f(x,t)}"></span>. If the initial location is given by a certain density function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo stretchy="false">(</mo>
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<mo stretchy="false">)</mo>
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<annotation encoding="application/x-tex">{\displaystyle g(x)}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="g(x)"></span>, then the density at time <i>t</i> is the <a href="/wiki/Convolution" title="Convolution">convolution</a> of <i>g</i> and the normal probability density function.</li></ul>
<h3><span class="mw-headline" id="Approximate_normality">Approximate normality</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=49" title="Edit section: Approximate normality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
<p><i>Approximately</i> normal distributions occur in many situations, as explained by the <a href="/wiki/Central_limit_theorem" title="Central limit theorem">central limit theorem</a>. When the outcome is produced by many small effects acting <i>additively and independently</i>, its distribution will be close to normal. The normal approximation will not be valid if the effects act multiplicatively (instead of additively), or if there is a single external influence that has a considerably larger magnitude than the rest of the effects.
</p>
<ul><li>In counting problems, where the central limit theorem includes a discrete-to-continuum approximation and where <a href="/wiki/Infinite_divisibility" title="Infinite divisibility">infinitely divisible</a> and <a href="/wiki/Indecomposable_distribution" title="Indecomposable distribution">decomposable</a> distributions are involved, such as
<ul><li><a href="/wiki/Binomial_distribution" title="Binomial distribution">Binomial random variables</a>, associated with binary response variables;</li>
<li><a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson random variables</a>, associated with rare events;</li></ul></li>
<li><a href="/wiki/Thermal_radiation" title="Thermal radiation">Thermal radiation</a> has a <a href="/wiki/Bose%E2%80%93Einstein_statistics" title="BoseEinstein statistics">BoseEinstein</a> distribution on very short time scales, and a normal distribution on longer timescales due to the central limit theorem.</li></ul>
<h3><span class="mw-headline" id="Assumed_normality">Assumed normality</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=50" title="Edit section: Assumed normality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
<figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Fisher_iris_versicolor_sepalwidth.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/Fisher_iris_versicolor_sepalwidth.svg/220px-Fisher_iris_versicolor_sepalwidth.svg.png" decoding="async" width="220" height="152" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/Fisher_iris_versicolor_sepalwidth.svg/330px-Fisher_iris_versicolor_sepalwidth.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/40/Fisher_iris_versicolor_sepalwidth.svg/440px-Fisher_iris_versicolor_sepalwidth.svg.png 2x" data-file-width="822" data-file-height="567" /></a><figcaption>Histogram of sepal widths for <i>Iris versicolor</i> from Fisher's <a href="/wiki/Iris_flower_data_set" title="Iris flower data set">Iris flower data set</a>, with superimposed best-fitting normal distribution.</figcaption></figure>
<style data-mw-deduplicate="TemplateStyles:r996844942">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 40px}.mw-parser-output .templatequote .templatequotecite{line-height:1.5em;text-align:left;padding-left:1.6em;margin-top:0}</style><blockquote class="templatequote"><p>I can only recognize the occurrence of the normal curve the Laplacian curve of errors as a very abnormal phenomenon. It is roughly approximated to in certain distributions; for this reason, and on account for its beautiful simplicity, we may, perhaps, use it as a first approximation, particularly in theoretical investigations.</p><div class="templatequotecite">—&#8202;<cite><a href="#CITEREFPearson1901">Pearson (1901)</a></cite></div></blockquote>
<p>There are statistical methods to empirically test that assumption; see the above <a href="#Normality_tests">Normality tests</a> section.
</p>
<ul><li>In <a href="/wiki/Biology" title="Biology">biology</a>, the <i>logarithm</i> of various variables tend to have a normal distribution, that is, they tend to have a <a href="/wiki/Log-normal_distribution" title="Log-normal distribution">log-normal distribution</a> (after separation on male/female subpopulations), with examples including:
<ul><li>Measures of size of living tissue (length, height, skin area, weight);<sup id="cite_ref-50" class="reference"><a href="#cite_note-50">&#91;50&#93;</a></sup></li>
<li>The <i>length</i> of <i>inert</i> appendages (hair, claws, nails, teeth) of biological specimens, <i>in the direction of growth</i>; presumably the thickness of tree bark also falls under this category;</li>
<li>Certain physiological measurements, such as blood pressure of adult humans.</li></ul></li>
<li>In finance, in particular the <a href="/wiki/Black%E2%80%93Scholes_model" title="BlackScholes model">BlackScholes model</a>, changes in the <i>logarithm</i> of exchange rates, price indices, and stock market indices are assumed normal (these variables behave like <a href="/wiki/Compound_interest" title="Compound interest">compound interest</a>, not like simple interest, and so are multiplicative). Some mathematicians such as <a href="/wiki/Benoit_Mandelbrot" title="Benoit Mandelbrot">Benoit Mandelbrot</a> have argued that <a href="/wiki/Levy_skew_alpha-stable_distribution" class="mw-redirect" title="Levy skew alpha-stable distribution">log-Levy distributions</a>, which possesses <a href="/wiki/Heavy_tails" class="mw-redirect" title="Heavy tails">heavy tails</a> would be a more appropriate model, in particular for the analysis for <a href="/wiki/Stock_market_crash" title="Stock market crash">stock market crashes</a>. The use of the assumption of normal distribution occurring in financial models has also been criticized by <a href="/wiki/Nassim_Nicholas_Taleb" title="Nassim Nicholas Taleb">Nassim Nicholas Taleb</a> in his works.</li>
<li><a href="/wiki/Propagation_of_uncertainty" title="Propagation of uncertainty">Measurement errors</a> in physical experiments are often modeled by a normal distribution. This use of a normal distribution does not imply that one is assuming the measurement errors are normally distributed, rather using the normal distribution produces the most conservative predictions possible given only knowledge about the mean and variance of the errors.<sup id="cite_ref-51" class="reference"><a href="#cite_note-51">&#91;51&#93;</a></sup></li>
<li>In <a href="/wiki/Standardized_testing_(statistics)" class="mw-redirect" title="Standardized testing (statistics)">standardized testing</a>, results can be made to have a normal distribution by either selecting the number and difficulty of questions (as in the <a href="/wiki/Intelligence_quotient" title="Intelligence quotient">IQ test</a>) or transforming the raw test scores into output scores by fitting them to the normal distribution. For example, the <a href="/wiki/SAT" title="SAT">SAT</a>'s traditional range of 200800 is based on a normal distribution with a mean of 500 and a standard deviation of 100.</li></ul>
<figure typeof="mw:File/Thumb"><a href="/wiki/File:FitNormDistr.tif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/FitNormDistr.tif/lossless-page1-220px-FitNormDistr.tif.png" decoding="async" width="220" height="164" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/FitNormDistr.tif/lossless-page1-330px-FitNormDistr.tif.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d8/FitNormDistr.tif/lossless-page1-440px-FitNormDistr.tif.png 2x" data-file-width="623" data-file-height="465" /></a><figcaption>Fitted cumulative normal distribution to October rainfalls, see <a href="/wiki/Distribution_fitting" class="mw-redirect" title="Distribution fitting">distribution fitting</a></figcaption></figure>
<ul><li>Many scores are derived from the normal distribution, including <a href="/wiki/Percentile_rank" title="Percentile rank">percentile ranks</a> (percentiles or quantiles), <a href="/wiki/Normal_curve_equivalent" title="Normal curve equivalent">normal curve equivalents</a>, <a href="/wiki/Stanine" title="Stanine">stanines</a>, <a href="/wiki/Standard_score" title="Standard score">z-scores</a>, and T-scores. Additionally, some behavioral statistical procedures assume that scores are normally distributed; for example, <a href="/wiki/Student%27s_t-test" title="Student&#39;s t-test">t-tests</a> and <a href="/wiki/Analysis_of_variance" title="Analysis of variance">ANOVAs</a>. <a href="/wiki/Bell_curve_grading" class="mw-redirect" title="Bell curve grading">Bell curve grading</a> assigns relative grades based on a normal distribution of scores.</li>
<li>In <a href="/wiki/Hydrology" title="Hydrology">hydrology</a> the distribution of long duration river discharge or rainfall, e.g. monthly and yearly totals, is often thought to be practically normal according to the <a href="/wiki/Central_limit_theorem" title="Central limit theorem">central limit theorem</a>.<sup id="cite_ref-52" class="reference"><a href="#cite_note-52">&#91;52&#93;</a></sup> The blue picture, made with <a href="/wiki/CumFreq" title="CumFreq">CumFreq</a>, illustrates an example of fitting the normal distribution to ranked October rainfalls showing the 90% <a href="/wiki/Confidence_belt" class="mw-redirect" title="Confidence belt">confidence belt</a> based on the <a href="/wiki/Binomial_distribution" title="Binomial distribution">binomial distribution</a>. The rainfall data are represented by <a href="/wiki/Plotting_position" class="mw-redirect" title="Plotting position">plotting positions</a> as part of the <a href="/wiki/Cumulative_frequency_analysis" title="Cumulative frequency analysis">cumulative frequency analysis</a>.</li></ul>
<h3><span class="mw-headline" id="Methodological_problems_and_peer_review">Methodological problems and peer review</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=51" title="Edit section: Methodological problems and peer review"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
<p><a href="/wiki/John_Ioannidis" title="John Ioannidis">John Ioannidis</a> argues that using normally distributed standard deviations as standards for validating research findings leave <a href="/wiki/Falsifiability" title="Falsifiability">falsifiable predictions</a> about phenomena that are not normally distributed untested. This includes, for example, phenomena that only appear when all necessary conditions are present and one cannot be a substitute for another in an addition-like way and phenomena that are not randomly distributed. Ioannidis argues that standard deviation-centered validation gives a false appearance of validity to hypotheses and theories where some but not all falsifiable predictions are normally distributed since the portion of falsifiable predictions that there is evidence against may and in some cases are in the non-normally distributed parts of the range of falsifiable predictions, as well as baselessly dismissing hypotheses for which none of the falsifiable predictions are normally distributed as if were they unfalsifiable when in fact they do make falsifiable predictions. It is argued by Ioannidis that many cases of mutually exclusive theories being accepted as validated by research journals are caused by failure of the journals to take in empirical falsifications of non-normally distributed predictions, and not because mutually exclusive theories are true, which they cannot be, although two mutually exclusive theories can both be wrong and a third one correct.<sup id="cite_ref-53" class="reference"><a href="#cite_note-53">&#91;53&#93;</a></sup>
</p>
<h2><span class="mw-headline" id="Computational_methods">Computational methods</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=52" title="Edit section: Computational methods"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h2>
<h3><span class="mw-headline" id="Generating_values_from_normal_distribution">Generating values from normal distribution</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=53" title="Edit section: Generating values from normal distribution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
<figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Planche_de_Galton.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Planche_de_Galton.jpg/250px-Planche_de_Galton.jpg" decoding="async" width="250" height="207" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Planche_de_Galton.jpg/375px-Planche_de_Galton.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/21/Planche_de_Galton.jpg/500px-Planche_de_Galton.jpg 2x" data-file-width="1350" data-file-height="1118" /></a><figcaption>The <a href="/wiki/Bean_machine" class="mw-redirect" title="Bean machine">bean machine</a>, a device invented by <a href="/wiki/Francis_Galton" title="Francis Galton">Francis Galton</a>, can be called the first generator of normal random variables. This machine consists of a vertical board with interleaved rows of pins. Small balls are dropped from the top and then bounce randomly left or right as they hit the pins. The balls are collected into bins at the bottom and settle down into a pattern resembling the Gaussian curve.</figcaption></figure>
<p>In computer simulations, especially in applications of the <a href="/wiki/Monte-Carlo_method" class="mw-redirect" title="Monte-Carlo method">Monte-Carlo method</a>, it is often desirable to generate values that are normally distributed. The algorithms listed below all generate the standard normal deviates, since a <span class="texhtml"><i>N</i>(<i>μ</i>, <i>σ</i><sup>2</sup>)</span> can be generated as <span class="texhtml"><i>X</i> = <i>μ</i> + <i>σZ</i></span>, where <i>Z</i> is standard normal. All these algorithms rely on the availability of a <a href="/wiki/Random_number_generator" class="mw-redirect" title="Random number generator">random number generator</a> <i>U</i> capable of producing <a href="/wiki/Uniform_distribution_(continuous)" class="mw-redirect" title="Uniform distribution (continuous)">uniform</a> random variates.
</p>
<ul><li>The most straightforward method is based on the <a href="/wiki/Probability_integral_transform" title="Probability integral transform">probability integral transform</a> property: if <i>U</i> is distributed uniformly on (0,1), then Φ<sup>1</sup>(<i>U</i>) will have the standard normal distribution. The drawback of this method is that it relies on calculation of the <a href="/wiki/Probit_function" class="mw-redirect" title="Probit function">probit function</a> Φ<sup>1</sup>, which cannot be done analytically. Some approximate methods are described in <a href="#CITEREFHart1968">Hart (1968)</a> and in the <a href="/wiki/Error_function" title="Error function">erf</a> article. Wichura gives a fast algorithm for computing this function to 16 decimal places,<sup id="cite_ref-54" class="reference"><a href="#cite_note-54">&#91;54&#93;</a></sup> which is used by <a href="/wiki/R_programming_language" class="mw-redirect" title="R programming language">R</a> to compute random variates of the normal distribution.</li>
<li><a href="/wiki/Irwin%E2%80%93Hall_distribution#Approximating_a_Normal_distribution" title="IrwinHall distribution">An easy-to-program approximate approach</a> that relies on the <a href="/wiki/Central_limit_theorem" title="Central limit theorem">central limit theorem</a> is as follows: generate 12 uniform <i>U</i>(0,1) deviates, add them all up, and subtract 6 the resulting random variable will have approximately standard normal distribution. In truth, the distribution will be <a href="/wiki/Irwin%E2%80%93Hall_distribution" title="IrwinHall distribution">IrwinHall</a>, which is a 12-section eleventh-order polynomial approximation to the normal distribution. This random deviate will have a limited range of (6,&#160;6).<sup id="cite_ref-55" class="reference"><a href="#cite_note-55">&#91;55&#93;</a></sup> Note that in a true normal distribution, only 0.00034% of all samples will fall outside ±6σ.</li>
<li>The <a href="/wiki/Box%E2%80%93Muller_transform" title="BoxMuller transform">BoxMuller method</a> uses two independent random numbers <i>U</i> and <i>V</i> distributed <a href="/wiki/Uniform_distribution_(continuous)" class="mw-redirect" title="Uniform distribution (continuous)">uniformly</a> on (0,1). Then the two random variables <i>X</i> and <i>Y</i> <div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X={\sqrt {-2\ln U}}\,\cos(2\pi V),\qquad Y={\sqrt {-2\ln U}}\,\sin(2\pi V).}">
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</math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51fa20f18a8a5ed19c147db4686e7b15b6ca2e38" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:54.779ex; height:3.343ex;" alt="{\displaystyle X={\sqrt {-2\ln U}}\,\cos(2\pi V),\qquad Y={\sqrt {-2\ln U}}\,\sin(2\pi V).}"></div> will both have the standard normal distribution, and will be <a href="/wiki/Independence_(probability_theory)" title="Independence (probability theory)">independent</a>. This formulation arises because for a <a href="/wiki/Bivariate_normal" class="mw-redirect" title="Bivariate normal">bivariate normal</a> random vector (<i>X</i>, <i>Y</i>) the squared norm <span class="texhtml"><i>X</i><sup>2</sup> + <i>Y</i><sup>2</sup></span> will have the <a href="/wiki/Chi-squared_distribution" title="Chi-squared distribution">chi-squared distribution</a> with two degrees of freedom, which is an easily generated <a href="/wiki/Exponential_distribution" title="Exponential distribution">exponential random variable</a> corresponding to the quantity 2&#160;ln(<i>U</i>) in these equations; and the angle is distributed uniformly around the circle, chosen by the random variable <i>V</i>.</li>
<li>The <a href="/wiki/Marsaglia_polar_method" title="Marsaglia polar method">Marsaglia polar method</a> is a modification of the BoxMuller method which does not require computation of the sine and cosine functions. In this method, <i>U</i> and <i>V</i> are drawn from the uniform (1,1) distribution, and then <span class="texhtml"><i>S</i> = <i>U</i><sup>2</sup> + <i>V</i><sup>2</sup></span> is computed. If <i>S</i> is greater or equal to 1, then the method starts over, otherwise the two quantities <div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=U{\sqrt {\frac {-2\ln S}{S}}},\qquad Y=V{\sqrt {\frac {-2\ln S}{S}}}}">
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</math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdace1879c7c786ba946a60e5acb29f354d86796" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:39.886ex; height:6.176ex;" alt="{\displaystyle X=U{\sqrt {\frac {-2\ln S}{S}}},\qquad Y=V{\sqrt {\frac {-2\ln S}{S}}}}"></div> are returned. Again, <i>X</i> and <i>Y</i> are independent, standard normal random variables.</li>
<li>The Ratio method<sup id="cite_ref-56" class="reference"><a href="#cite_note-56">&#91;56&#93;</a></sup> is a rejection method. The algorithm proceeds as follows:
<ul><li>Generate two independent uniform deviates <i>U</i> and <i>V</i>;</li>
<li>Compute <i>X</i> = <span class="nowrap">&#8730;<span style="border-top:1px solid; padding:0 0.1em;">8/<i>e</i></span></span> (<i>V</i> 0.5)/<i>U</i>;</li>
<li>Optional: if <i>X</i><sup>2</sup> ≤ 5 4<i>e</i><sup>1/4</sup><i>U</i> then accept <i>X</i> and terminate algorithm;</li>
<li>Optional: if <i>X</i><sup>2</sup> ≥ 4<i>e</i><sup>1.35</sup>/<i>U</i> + 1.4 then reject <i>X</i> and start over from step 1;</li>
<li>If <i>X</i><sup>2</sup> ≤ 4 ln<i>U</i> then accept <i>X</i>, otherwise start over the algorithm.</li></ul>
<dl><dd>The two optional steps allow the evaluation of the logarithm in the last step to be avoided in most cases. These steps can be greatly improved<sup id="cite_ref-57" class="reference"><a href="#cite_note-57">&#91;57&#93;</a></sup> so that the logarithm is rarely evaluated.</dd></dl></li>
<li>The <a href="/wiki/Ziggurat_algorithm" title="Ziggurat algorithm">ziggurat algorithm</a><sup id="cite_ref-58" class="reference"><a href="#cite_note-58">&#91;58&#93;</a></sup> is faster than the BoxMuller transform and still exact. In about 97% of all cases it uses only two random numbers, one random integer and one random uniform, one multiplication and an if-test. Only in 3% of the cases, where the combination of those two falls outside the "core of the ziggurat" (a kind of rejection sampling using logarithms), do exponentials and more uniform random numbers have to be employed.</li>
<li>Integer arithmetic can be used to sample from the standard normal distribution.<sup id="cite_ref-59" class="reference"><a href="#cite_note-59">&#91;59&#93;</a></sup> This method is exact in the sense that it satisfies the conditions of <i>ideal approximation</i>;<sup id="cite_ref-60" class="reference"><a href="#cite_note-60">&#91;60&#93;</a></sup> i.e., it is equivalent to sampling a real number from the standard normal distribution and rounding this to the nearest representable floating point number.</li>
<li>There is also some investigation<sup id="cite_ref-61" class="reference"><a href="#cite_note-61">&#91;61&#93;</a></sup> into the connection between the fast <a href="/wiki/Hadamard_transform" title="Hadamard transform">Hadamard transform</a> and the normal distribution, since the transform employs just addition and subtraction and by the central limit theorem random numbers from almost any distribution will be transformed into the normal distribution. In this regard a series of Hadamard transforms can be combined with random permutations to turn arbitrary data sets into a normally distributed data.</li></ul>
<h3><span class="mw-headline" id="Numerical_approximations_for_the_normal_cumulative_distribution_function_and_normal_quantile_function">Numerical approximations for the normal cumulative distribution function and normal quantile function</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=54" title="Edit section: Numerical approximations for the normal cumulative distribution function and normal quantile function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
<p>The standard normal <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a> is widely used in scientific and statistical computing.
</p><p>The values Φ(<i>x</i>) may be approximated very accurately by a variety of methods, such as <a href="/wiki/Numerical_integration" title="Numerical integration">numerical integration</a>, <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a>, <a href="/wiki/Asymptotic_series" class="mw-redirect" title="Asymptotic series">asymptotic series</a> and <a href="/wiki/Gauss%27s_continued_fraction#Of_Kummer&#39;s_confluent_hypergeometric_function" title="Gauss&#39;s continued fraction">continued fractions</a>. Different approximations are used depending on the desired level of accuracy.
</p>
<ul><li><a href="#CITEREFZelenSevero1964">Zelen &amp; Severo (1964)</a> give the approximation for Φ(<i>x</i>) for <i>x</i> &gt; 0 with the absolute error <span class="texhtml">&#124;<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>ε</i>(<i>x</i>)</span>&#124; &lt; 7.5·10<sup>8</sup></span> (algorithm <a rel="nofollow" class="external text" href="https://secure.math.ubc.ca/~cbm/aands/page_932.htm">26.2.17</a>): <div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (x)=1-\varphi (x)\left(b_{1}t+b_{2}t^{2}+b_{3}t^{3}+b_{4}t^{4}+b_{5}t^{5}\right)+\varepsilon (x),\qquad t={\frac {1}{1+b_{0}x}},}">
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</math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202a295cd562d4d7404a1042e23f14b8d72be308" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:74.677ex; height:5.676ex;" alt="{\displaystyle \Phi (x)=1-\varphi (x)\left(b_{1}t+b_{2}t^{2}+b_{3}t^{3}+b_{4}t^{4}+b_{5}t^{5}\right)+\varepsilon (x),\qquad t={\frac {1}{1+b_{0}x}},}"></div> where <i>ϕ</i>(<i>x</i>) is the standard normal probability density function, and <i>b</i><sub>0</sub> = 0.2316419, <i>b</i><sub>1</sub> = 0.319381530, <i>b</i><sub>2</sub> = 0.356563782, <i>b</i><sub>3</sub> = 1.781477937, <i>b</i><sub>4</sub> = 1.821255978, <i>b</i><sub>5</sub> = 1.330274429.</li>
<li><a href="#CITEREFHart1968">Hart (1968)</a> lists some dozens of approximations by means of rational functions, with or without exponentials for the <style data-mw-deduplicate="TemplateStyles:r886049734">.mw-parser-output .monospaced{font-family:monospace,monospace}</style><span class="monospaced">erfc()</span> function. His algorithms vary in the degree of complexity and the resulting precision, with maximum absolute precision of 24 digits. An algorithm by <a href="#CITEREFWest2009">West (2009)</a> combines Hart's algorithm 5666 with a <a href="/wiki/Continued_fraction" title="Continued fraction">continued fraction</a> approximation in the tail to provide a fast computation algorithm with a 16-digit precision.</li>
<li><a href="#CITEREFCody1969">Cody (1969)</a> after recalling Hart68 solution is not suited for erf, gives a solution for both erf and erfc, with maximal relative error bound, via <a href="/wiki/Rational_function" title="Rational function">Rational Chebyshev Approximation</a>.</li>
<li><a href="#CITEREFMarsaglia2004">Marsaglia (2004)</a> suggested a simple algorithm<sup id="cite_ref-62" class="reference"><a href="#cite_note-62">&#91;note 1&#93;</a></sup> based on the Taylor series expansion <div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (x)={\frac {1}{2}}+\varphi (x)\left(x+{\frac {x^{3}}{3}}+{\frac {x^{5}}{3\cdot 5}}+{\frac {x^{7}}{3\cdot 5\cdot 7}}+{\frac {x^{9}}{3\cdot 5\cdot 7\cdot 9}}+\cdots \right)}">
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<li>The <a href="/wiki/GNU_Scientific_Library" title="GNU Scientific Library">GNU Scientific Library</a> calculates values of the standard normal cumulative distribution function using Hart's algorithms and approximations with <a href="/wiki/Chebyshev_polynomial" class="mw-redirect" title="Chebyshev polynomial">Chebyshev polynomials</a>.</li>
<li><a href="#CITEREFDia2023">Dia (2023)</a> proposes the following approximation of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-\Phi }">
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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}1-\Phi \left(x\right)&amp;=\left({\frac {0.39894228040143268}{x+2.92678600515804815}}\right)\left({\frac {x^{2}+8.42742300458043240x+18.38871225773938487}{x^{2}+5.81582518933527391x+8.97280659046817350}}\right)\\&amp;\left({\frac {x^{2}+7.30756258553673541x+18.25323235347346525}{x^{2}+5.70347935898051437x+10.27157061171363079}}\right)\left({\frac {x^{2}+5.66479518878470765x+18.61193318971775795}{x^{2}+5.51862483025707963x+12.72323261907760928}}\right)\\&amp;\left({\frac {x^{2}+4.91396098895240075x+24.14804072812762821}{x^{2}+5.26184239579604207x+16.88639562007936908}}\right)\left({\frac {x^{2}+3.83362947800146179x+11.61511226260603247}{x^{2}+4.92081346632882033x+24.12333774572479110}}\right)e^{-{\frac {x^{2}}{2}}}\end{aligned}}}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/959a81d76abcc98fb985f19743f769c89769085a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -9.005ex; width:131.64ex; height:19.176ex;" alt="{\displaystyle {\begin{aligned}1-\Phi \left(x\right)&amp;=\left({\frac {0.39894228040143268}{x+2.92678600515804815}}\right)\left({\frac {x^{2}+8.42742300458043240x+18.38871225773938487}{x^{2}+5.81582518933527391x+8.97280659046817350}}\right)\\&amp;\left({\frac {x^{2}+7.30756258553673541x+18.25323235347346525}{x^{2}+5.70347935898051437x+10.27157061171363079}}\right)\left({\frac {x^{2}+5.66479518878470765x+18.61193318971775795}{x^{2}+5.51862483025707963x+12.72323261907760928}}\right)\\&amp;\left({\frac {x^{2}+4.91396098895240075x+24.14804072812762821}{x^{2}+5.26184239579604207x+16.88639562007936908}}\right)\left({\frac {x^{2}+3.83362947800146179x+11.61511226260603247}{x^{2}+4.92081346632882033x+24.12333774572479110}}\right)e^{-{\frac {x^{2}}{2}}}\end{aligned}}}"></span> and for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x&lt;0}">
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<annotation encoding="application/x-tex">{\displaystyle x&lt;0}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a4dbbf970b2d2863dcab589eafe006f08e727d7" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x&lt;0}"></span>,</li></ul>
<p><div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-\Phi \left(x\right)=1-\left(1-\Phi \left(-x\right)\right)}">
<semantics>
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<annotation encoding="application/x-tex">{\displaystyle 1-\Phi \left(x\right)=1-\left(1-\Phi \left(-x\right)\right)}</annotation>
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</math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e12b409099845b3057fa7bdb2d9b84c6cacf73a" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.133ex; height:2.843ex;" alt="{\displaystyle 1-\Phi \left(x\right)=1-\left(1-\Phi \left(-x\right)\right)}"></div>
</p><p>Shore (1982) introduced simple approximations that may be incorporated in stochastic optimization models of engineering and operations research, like reliability engineering and inventory analysis. Denoting <span class="texhtml"><i>p</i> = Φ(<i>z</i>)</span>, the simplest approximation for the quantile function is:
<div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=\Phi ^{-1}(p)=5.5556\left[1-\left({\frac {1-p}{p}}\right)^{0.1186}\right],\qquad p\geq 1/2}">
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<annotation encoding="application/x-tex">{\displaystyle z=\Phi ^{-1}(p)=5.5556\left[1-\left({\frac {1-p}{p}}\right)^{0.1186}\right],\qquad p\geq 1/2}</annotation>
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</math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f2df7f1427d0c90d075faef38f4f5ab7acce5c9" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:55.885ex; height:7.509ex;" alt="{\displaystyle z=\Phi ^{-1}(p)=5.5556\left[1-\left({\frac {1-p}{p}}\right)^{0.1186}\right],\qquad p\geq 1/2}"></div>
</p><p>This approximation delivers for <i>z</i> a maximum absolute error of 0.026 (for <span class="texhtml">0.5 ≤ <i>p</i> ≤ 0.9999</span>, corresponding to <span class="texhtml">0 ≤ <i>z</i> ≤ 3.719</span>). For <span class="texhtml"><i>p</i> &lt; 1/2</span> replace <i>p</i> by <span class="texhtml">1 <i>p</i></span> and change sign. Another approximation, somewhat less accurate, is the single-parameter approximation:
<div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=-0.4115\left\{{\frac {1-p}{p}}+\log \left[{\frac {1-p}{p}}\right]-1\right\},\qquad p\geq 1/2}">
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<annotation encoding="application/x-tex">{\displaystyle z=-0.4115\left\{{\frac {1-p}{p}}+\log \left[{\frac {1-p}{p}}\right]-1\right\},\qquad p\geq 1/2}</annotation>
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</math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1edea9f990058f741db6735799c8b40999b833b" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:54.435ex; height:6.176ex;" alt="{\displaystyle z=-0.4115\left\{{\frac {1-p}{p}}+\log \left[{\frac {1-p}{p}}\right]-1\right\},\qquad p\geq 1/2}"></div>
</p><p>The latter had served to derive a simple approximation for the loss integral of the normal distribution, defined by
<div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}L(z)&amp;=\int _{z}^{\infty }(u-z)\varphi (u)\,du=\int _{z}^{\infty }[1-\Phi (u)]\,du\\[5pt]L(z)&amp;\approx {\begin{cases}0.4115\left({\dfrac {p}{1-p}}\right)-z,&amp;p&lt;1/2,\\\\0.4115\left({\dfrac {1-p}{p}}\right),&amp;p\geq 1/2.\end{cases}}\\[5pt]{\text{or, equivalently,}}\\L(z)&amp;\approx {\begin{cases}0.4115\left\{1-\log \left[{\frac {p}{1-p}}\right]\right\},&amp;p&lt;1/2,\\\\0.4115{\dfrac {1-p}{p}},&amp;p\geq 1/2.\end{cases}}\end{aligned}}}">
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<mtext>or, equivalently,</mtext>
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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}L(z)&amp;=\int _{z}^{\infty }(u-z)\varphi (u)\,du=\int _{z}^{\infty }[1-\Phi (u)]\,du\\[5pt]L(z)&amp;\approx {\begin{cases}0.4115\left({\dfrac {p}{1-p}}\right)-z,&amp;p&lt;1/2,\\\\0.4115\left({\dfrac {1-p}{p}}\right),&amp;p\geq 1/2.\end{cases}}\\[5pt]{\text{or, equivalently,}}\\L(z)&amp;\approx {\begin{cases}0.4115\left\{1-\log \left[{\frac {p}{1-p}}\right]\right\},&amp;p&lt;1/2,\\\\0.4115{\dfrac {1-p}{p}},&amp;p\geq 1/2.\end{cases}}\end{aligned}}}</annotation>
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</math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4b69fa586cffdfbbd40a94c65629726e4ae78bf" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -19.338ex; width:59.621ex; height:39.843ex;" alt="{\displaystyle {\begin{aligned}L(z)&amp;=\int _{z}^{\infty }(u-z)\varphi (u)\,du=\int _{z}^{\infty }[1-\Phi (u)]\,du\\[5pt]L(z)&amp;\approx {\begin{cases}0.4115\left({\dfrac {p}{1-p}}\right)-z,&amp;p&lt;1/2,\\\\0.4115\left({\dfrac {1-p}{p}}\right),&amp;p\geq 1/2.\end{cases}}\\[5pt]{\text{or, equivalently,}}\\L(z)&amp;\approx {\begin{cases}0.4115\left\{1-\log \left[{\frac {p}{1-p}}\right]\right\},&amp;p&lt;1/2,\\\\0.4115{\dfrac {1-p}{p}},&amp;p\geq 1/2.\end{cases}}\end{aligned}}}"></div>
</p><p>This approximation is particularly accurate for the right far-tail (maximum error of 10<sup>3</sup> for z≥1.4). Highly accurate approximations for the cumulative distribution function, based on <a href="/wiki/Response_modeling_methodology" title="Response modeling methodology">Response Modeling Methodology</a> (RMM, Shore, 2011, 2012), are shown in Shore (2005).
</p><p>Some more approximations can be found at: <a href="/wiki/Error_function#Approximation_with_elementary_functions" title="Error function">Error function#Approximation with elementary functions</a>. In particular, small <i>relative</i> error on the whole domain for the cumulative distribution function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }">
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<mi mathvariant="normal">&#x03A6;<!-- Φ --></mi>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="\Phi "></span> and the quantile function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi ^{-1}}">
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab22c7cf7f1a54d85993e0257a93f28eae546df8" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.011ex; height:2.676ex;" alt="\Phi ^{-1}"></span> as well, is achieved via an explicitly invertible formula by Sergei Winitzki in 2008.
</p>
<h2><span class="mw-headline" id="History">History</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=55" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h2>
<h3><span class="mw-headline" id="Development">Development</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=56" title="Edit section: Development"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
<p>Some authors<sup id="cite_ref-63" class="reference"><a href="#cite_note-63">&#91;62&#93;</a></sup><sup id="cite_ref-64" class="reference"><a href="#cite_note-64">&#91;63&#93;</a></sup> attribute the credit for the discovery of the normal distribution to <a href="/wiki/Abraham_de_Moivre" title="Abraham de Moivre">de Moivre</a>, who in 1738<sup id="cite_ref-65" class="reference"><a href="#cite_note-65">&#91;note 2&#93;</a></sup> published in the second edition of his <i><a href="/wiki/The_Doctrine_of_Chances" title="The Doctrine of Chances">The Doctrine of Chances</a></i> the study of the coefficients in the <a href="/wiki/Binomial_expansion" class="mw-redirect" title="Binomial expansion">binomial expansion</a> of <span class="texhtml">(<i>a</i> + <i>b</i>)<sup><i>n</i></sup></span>. De Moivre proved that the middle term in this expansion has the approximate magnitude of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 2^{n}/{\sqrt {2\pi n}}}">
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5327ed8841e09d62970ee806553294cdfe96e9e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.838ex; height:3.509ex;" alt="{\textstyle -{\frac {2\ell \ell }{n}}}"></span>."<sup id="cite_ref-66" class="reference"><a href="#cite_note-66">&#91;64&#93;</a></sup> Although this theorem can be interpreted as the first obscure expression for the normal probability law, <a href="/wiki/Stephen_Stigler" title="Stephen Stigler">Stigler</a> points out that de Moivre himself did not interpret his results as anything more than the approximate rule for the binomial coefficients, and in particular de Moivre lacked the concept of the probability density function.<sup id="cite_ref-67" class="reference"><a href="#cite_note-67">&#91;65&#93;</a></sup>
</p>
<figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Carl_Friedrich_Gauss.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Carl_Friedrich_Gauss.jpg/180px-Carl_Friedrich_Gauss.jpg" decoding="async" width="180" height="232" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Carl_Friedrich_Gauss.jpg/270px-Carl_Friedrich_Gauss.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Carl_Friedrich_Gauss.jpg/360px-Carl_Friedrich_Gauss.jpg 2x" data-file-width="917" data-file-height="1180" /></a><figcaption><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a> discovered the normal distribution in 1809 as a way to rationalize the <a href="/wiki/Method_of_least_squares" class="mw-redirect" title="Method of least squares">method of least squares</a>.</figcaption></figure>
<p>In 1823 <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a> published his monograph <span title="Theory of the Combination of Observations Least Subject to Errors">"<i>Theoria combinationis observationum erroribus minimis obnoxiae</i>"</span> where among other things he introduces several important statistical concepts, such as the <a href="/wiki/Method_of_least_squares" class="mw-redirect" title="Method of least squares">method of least squares</a>, the <a href="/wiki/Method_of_maximum_likelihood" class="mw-redirect" title="Method of maximum likelihood">method of maximum likelihood</a>, and the <i>normal distribution</i>. Gauss used <i>M</i>, <span class="nowrap"><i>M</i></span>, <span class="nowrap"><i>M</i>, ...</span> to denote the measurements of some unknown quantity&#160;<i>V</i>, and sought the most probable estimator of that quantity: the one that maximizes the probability <span class="texhtml"><i>φ</i>(<i>M</i> <i>V</i>) · <i>φ</i>(<i>M</i> <i>V</i>) · <i>φ</i>(<i>M</i> <i>V</i>) · ...</span> of obtaining the observed experimental results. In his notation φΔ is the probability density function of the measurement errors of magnitude Δ. Not knowing what the function <i>φ</i> is, Gauss requires that his method should reduce to the well-known answer: the arithmetic mean of the measured values.<sup id="cite_ref-68" class="reference"><a href="#cite_note-68">&#91;note 3&#93;</a></sup> Starting from these principles, Gauss demonstrates that the only law that rationalizes the choice of arithmetic mean as an estimator of the location parameter, is the normal law of errors:<sup id="cite_ref-69" class="reference"><a href="#cite_note-69">&#91;66&#93;</a></sup>
<div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi {\mathit {\Delta }}={\frac {h}{\surd \pi }}\,e^{-\mathrm {hh} \Delta \Delta },}">
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where <i>h</i> is "the measure of the precision of the observations". Using this normal law as a generic model for errors in the experiments, Gauss formulates what is now known as the <a href="/wiki/Non-linear_least_squares" title="Non-linear least squares">non-linear</a> <a href="/wiki/Weighted_least_squares" title="Weighted least squares">weighted least squares</a> method.<sup id="cite_ref-70" class="reference"><a href="#cite_note-70">&#91;67&#93;</a></sup>
</p>
<figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Pierre-Simon_Laplace.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Pierre-Simon_Laplace.jpg/180px-Pierre-Simon_Laplace.jpg" decoding="async" width="180" height="196" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Pierre-Simon_Laplace.jpg/270px-Pierre-Simon_Laplace.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Pierre-Simon_Laplace.jpg/360px-Pierre-Simon_Laplace.jpg 2x" data-file-width="550" data-file-height="600" /></a><figcaption><a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Pierre-Simon Laplace</a> proved the <a href="/wiki/Central_limit_theorem" title="Central limit theorem">central limit theorem</a> in 1810, consolidating the importance of the normal distribution in statistics.</figcaption></figure>
<p>Although Gauss was the first to suggest the normal distribution law, <a href="/wiki/Pierre_Simon_de_Laplace" class="mw-redirect" title="Pierre Simon de Laplace">Laplace</a> made significant contributions.<sup id="cite_ref-71" class="reference"><a href="#cite_note-71">&#91;note 4&#93;</a></sup> It was Laplace who first posed the problem of aggregating several observations in 1774,<sup id="cite_ref-72" class="reference"><a href="#cite_note-72">&#91;68&#93;</a></sup> although his own solution led to the <a href="/wiki/Laplacian_distribution" class="mw-redirect" title="Laplacian distribution">Laplacian distribution</a>. It was Laplace who first calculated the value of the <a href="/wiki/Gaussian_integral" title="Gaussian integral">integral <span class="nowrap">∫ <i>e</i><sup><i>t</i><sup>2</sup></sup>&#160;<i>dt</i> = <span class="nowrap">&#8730;<span style="border-top:1px solid; padding:0 0.1em;"><span class="texhtml mvar" style="font-style:italic;">π</span></span></span></span></a> in 1782, providing the normalization constant for the normal distribution.<sup id="cite_ref-73" class="reference"><a href="#cite_note-73">&#91;69&#93;</a></sup> Finally, it was Laplace who in 1810 proved and presented to the Academy the fundamental <a href="/wiki/Central_limit_theorem" title="Central limit theorem">central limit theorem</a>, which emphasized the theoretical importance of the normal distribution.<sup id="cite_ref-74" class="reference"><a href="#cite_note-74">&#91;70&#93;</a></sup>
</p><p>It is of interest to note that in 1809 an Irish-American mathematician <a href="/wiki/Robert_Adrain" title="Robert Adrain">Robert Adrain</a> published two insightful but flawed derivations of the normal probability law, simultaneously and independently from Gauss.<sup id="cite_ref-75" class="reference"><a href="#cite_note-75">&#91;71&#93;</a></sup> His works remained largely unnoticed by the scientific community, until in 1871 they were exhumed by <a href="/wiki/Cleveland_Abbe" title="Cleveland Abbe">Abbe</a>.<sup id="cite_ref-76" class="reference"><a href="#cite_note-76">&#91;72&#93;</a></sup>
</p><p>In the middle of the 19th century <a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell</a> demonstrated that the normal distribution is not just a convenient mathematical tool, but may also occur in natural phenomena:<sup id="cite_ref-77" class="reference"><a href="#cite_note-77">&#91;73&#93;</a></sup> The number of particles whose velocity, resolved in a certain direction, lies between <i>x</i> and <i>x</i>&#160;+&#160;<i>dx</i> is
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</p>
<h3><span class="mw-headline" id="Naming">Naming</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=57" title="Edit section: Naming"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
<p>Today, the concept is usually known in English as the <b>normal distribution</b> or <b>Gaussian distribution</b>. Other less common names include Gauss distribution, Laplace-Gauss distribution, the law of error, the law of facility of errors, Laplace's second law, and Gaussian law.
</p><p>Gauss himself apparently coined the term with reference to the "normal equations" involved in its applications, with normal having its technical meaning of orthogonal rather than usual.<sup id="cite_ref-78" class="reference"><a href="#cite_note-78">&#91;74&#93;</a></sup> However, by the end of the 19th century some authors<sup id="cite_ref-79" class="reference"><a href="#cite_note-79">&#91;note 5&#93;</a></sup> had started using the name <i>normal distribution</i>, where the word "normal" was used as an adjective&#160; the term now being seen as a reflection of the fact that this distribution was seen as typical, common&#160; and thus normal. <a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Peirce</a> (one of those authors) once defined "normal" thus: "...the 'normal' is not the average (or any other kind of mean) of what actually occurs, but of what <i>would</i>, in the long run, occur under certain circumstances."<sup id="cite_ref-80" class="reference"><a href="#cite_note-80">&#91;75&#93;</a></sup> Around the turn of the 20th century <a href="/wiki/Karl_Pearson" title="Karl Pearson">Pearson</a> popularized the term <i>normal</i> as a designation for this distribution.<sup id="cite_ref-81" class="reference"><a href="#cite_note-81">&#91;76&#93;</a></sup>
</p>
<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r996844942"><blockquote class="templatequote"><p>Many years ago I called the LaplaceGaussian curve the <i>normal</i> curve, which name, while it avoids an international question of priority, has the disadvantage of leading people to believe that all other distributions of frequency are in one sense or another 'abnormal'. </p><div class="templatequotecite">—&#8202;<cite><a href="#CITEREFPearson1920">Pearson (1920)</a></cite></div></blockquote>
<p>Also, it was Pearson who first wrote the distribution in terms of the standard deviation <i>σ</i> as in modern notation. Soon after this, in year 1915, <a href="/wiki/Ronald_Fisher" title="Ronald Fisher">Fisher</a> added the location parameter to the formula for normal distribution, expressing it in the way it is written nowadays:
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</p><p>The term "standard normal", which denotes the normal distribution with zero mean and unit variance came into general use around the 1950s, appearing in the popular textbooks by P.&#160;G. Hoel (1947) <i>Introduction to Mathematical Statistics</i> and A.&#160;M. Mood (1950) <i>Introduction to the Theory of Statistics</i>.<sup id="cite_ref-82" class="reference"><a href="#cite_note-82">&#91;77&#93;</a></sup>
</p>
<h2><span class="mw-headline" id="See_also">See also</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=58" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h2>
<style data-mw-deduplicate="TemplateStyles:r1132942124">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:solid #aaa 1px;padding:0.1em;background:#f9f9f9}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{clear:left;float:left;margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright">
<li class="portalbox-entry"><span class="portalbox-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></span><span class="portalbox-link"><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></span></li></ul>
<ul><li><a href="/wiki/Bates_distribution" title="Bates distribution">Bates distribution</a> similar to the IrwinHall distribution, but rescaled back into the 0 to 1 range</li>
<li><a href="/wiki/Behrens%E2%80%93Fisher_problem" title="BehrensFisher problem">BehrensFisher problem</a> the long-standing problem of testing whether two normal samples with different variances have same means;</li>
<li><a href="/wiki/Bhattacharyya_distance" title="Bhattacharyya distance">Bhattacharyya distance</a> method used to separate mixtures of normal distributions</li>
<li><a href="/wiki/Erd%C5%91s%E2%80%93Kac_theorem" title="ErdősKac theorem">ErdősKac theorem</a> on the occurrence of the normal distribution in <a href="/wiki/Number_theory" title="Number theory">number theory</a></li>
<li><a href="/wiki/Full_width_at_half_maximum" title="Full width at half maximum">Full width at half maximum</a></li>
<li><a href="/wiki/Gaussian_blur" title="Gaussian blur">Gaussian blur</a> <a href="/wiki/Convolution" title="Convolution">convolution</a>, which uses the normal distribution as a kernel</li>
<li><a href="/wiki/Modified_half-normal_distribution" title="Modified half-normal distribution">Modified half-normal distribution</a><sup id="cite_ref-Sun,_Kong_and_Pal_83-0" class="reference"><a href="#cite_note-Sun,_Kong_and_Pal-83">&#91;78&#93;</a></sup> with the pdf on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,\infty )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>,</mo>
<mi mathvariant="normal">&#x221E;<!-- ∞ --></mi>
<mo stretchy="false">)</mo>
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<annotation encoding="application/x-tex">{\displaystyle (0,\infty )}</annotation>
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</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da17102e4ed0886686094ee531df040d2e86352a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.329ex; height:2.843ex;" alt="(0,\infty )"></span> is given as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\frac {2\beta ^{\frac {\alpha }{2}}x^{\alpha -1}\exp(-\beta x^{2}+\gamma x)}{\Psi {\left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
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<mo>&#x2061;<!-- --></mo>
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<mrow class="MJX-TeXAtom-ORD">
<mrow>
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<mi>&#x03B1;<!-- α --></mi>
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<annotation encoding="application/x-tex">{\displaystyle f(x)={\frac {2\beta ^{\frac {\alpha }{2}}x^{\alpha -1}\exp(-\beta x^{2}+\gamma x)}{\Psi {\left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba362b2bb9616f39c06cb4214bf4d8df1d14dc4e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:34.194ex; height:10.509ex;" alt="{\displaystyle f(x)={\frac {2\beta ^{\frac {\alpha }{2}}x^{\alpha -1}\exp(-\beta x^{2}+\gamma x)}{\Psi {\left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}}}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi (\alpha ,z)={}_{1}\Psi _{1}\left({\begin{matrix}\left(\alpha ,{\frac {1}{2}}\right)\\(1,0)\end{matrix}};z\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">&#x03A8;<!-- Ψ --></mi>
<mo stretchy="false">(</mo>
<mi>&#x03B1;<!-- α --></mi>
<mo>,</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msub>
<mrow class="MJX-TeXAtom-ORD">
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
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<msub>
<mi mathvariant="normal">&#x03A8;<!-- Ψ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
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<mo>(</mo>
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<mrow class="MJX-TeXAtom-ORD">
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>(</mo>
<mrow>
<mi>&#x03B1;<!-- α --></mi>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
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</mtr>
<mtr>
<mtd>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>,</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mtd>
</mtr>
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<mo>;</mo>
<mi>z</mi>
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<annotation encoding="application/x-tex">{\displaystyle \Psi (\alpha ,z)={}_{1}\Psi _{1}\left({\begin{matrix}\left(\alpha ,{\frac {1}{2}}\right)\\(1,0)\end{matrix}};z\right)}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f318f1c6f5b6c50886d35fe09b9205c3e66784" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.494ex; height:7.509ex;" alt="{\displaystyle \Psi (\alpha ,z)={}_{1}\Psi _{1}\left({\begin{matrix}\left(\alpha ,{\frac {1}{2}}\right)\\(1,0)\end{matrix}};z\right)}"></span> denotes the <a href="/wiki/Fox%E2%80%93Wright_Psi_function" class="mw-redirect" title="FoxWright Psi function">FoxWright Psi function</a>.</li>
<li><a href="/wiki/Normally_distributed_and_uncorrelated_does_not_imply_independent" title="Normally distributed and uncorrelated does not imply independent">Normally distributed and uncorrelated does not imply independent</a></li>
<li><a href="/wiki/Ratio_normal_distribution" class="mw-redirect" title="Ratio normal distribution">Ratio normal distribution</a></li>
<li><a href="/wiki/Reciprocal_normal_distribution" class="mw-redirect" title="Reciprocal normal distribution">Reciprocal normal distribution</a></li>
<li><a href="/wiki/Standard_normal_table" title="Standard normal table">Standard normal table</a></li>
<li><a href="/wiki/Stein%27s_lemma" title="Stein&#39;s lemma">Stein's lemma</a></li>
<li><a href="/wiki/Sub-Gaussian_distribution" title="Sub-Gaussian distribution">Sub-Gaussian distribution</a></li>
<li><a href="/wiki/Sum_of_normally_distributed_random_variables" title="Sum of normally distributed random variables">Sum of normally distributed random variables</a></li>
<li><a href="/wiki/Tweedie_distribution" title="Tweedie distribution">Tweedie distribution</a> The normal distribution is a member of the family of Tweedie <a href="/wiki/Exponential_dispersion_model" title="Exponential dispersion model">exponential dispersion models</a>.</li>
<li><a href="/wiki/Wrapped_normal_distribution" title="Wrapped normal distribution">Wrapped normal distribution</a> the Normal distribution applied to a circular domain</li>
<li><a href="/wiki/Z-test" title="Z-test">Z-test</a> using the normal distribution</li></ul>
<h2><span class="mw-headline" id="Notes">Notes</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=59" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h2>
<style data-mw-deduplicate="TemplateStyles:r1011085734">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist">
<div class="mw-references-wrap"><ol class="references">
<li id="cite_note-62"><span class="mw-cite-backlink"><b><a href="#cite_ref-62">^</a></b></span> <span class="reference-text">For example, this algorithm is given in the article <a href="/wiki/Bc_programming_language#A_translated_C_function" class="mw-redirect" title="Bc programming language">Bc programming language</a>.</span>
</li>
<li id="cite_note-65"><span class="mw-cite-backlink"><b><a href="#cite_ref-65">^</a></b></span> <span class="reference-text">De Moivre first published his findings in 1733, in a pamphlet <i>Approximatio ad Summam Terminorum Binomii <span class="texhtml">(</span></i>a<i> + </i>b<i>)<sup></sup></i>n<i></i></span><i> in Seriem Expansi</i> that was designated for private circulation only. But it was not until the year 1738 that he made his results publicly available. The original pamphlet was reprinted several times, see for example <a href="#CITEREFWalker1985">Walker (1985)</a>.
</li>
<li id="cite_note-68"><span class="mw-cite-backlink"><b><a href="#cite_ref-68">^</a></b></span> <span class="reference-text">"It has been customary certainly to regard as an axiom the hypothesis that if any quantity has been determined by several direct observations, made under the same circumstances and with equal care, the arithmetical mean of the observed values affords the most probable value, if not rigorously, yet very nearly at least, so that it is always most safe to adhere to it." — <a href="#CITEREFGauss1809">Gauss (1809</a>, section 177) </span>
</li>
<li id="cite_note-71"><span class="mw-cite-backlink"><b><a href="#cite_ref-71">^</a></b></span> <span class="reference-text">"My custom of terming the curve the GaussLaplacian or <i>normal</i> curve saves us from proportioning the merit of discovery between the two great astronomer mathematicians." quote from <a href="#CITEREFPearson1905">Pearson (1905</a>, p.&#160;189) </span>
</li>
<li id="cite_note-79"><span class="mw-cite-backlink"><b><a href="#cite_ref-79">^</a></b></span> <span class="reference-text">Besides those specifically referenced here, such use is encountered in the works of <a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Peirce</a>, <a href="/wiki/Francis_Galton" title="Francis Galton">Galton</a> (<a href="#CITEREFGalton1889">Galton (1889</a>, chapter V)) and <a href="/wiki/Wilhelm_Lexis" title="Wilhelm Lexis">Lexis</a> (<a href="#CITEREFLexis1878">Lexis (1878)</a>, <a href="#CITEREFRohrbasserVéron2003">Rohrbasser &amp; Véron (2003)</a>) c. 1875.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">&#91;<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (June 2011)">citation needed</span></a></i>&#93;</sup> </span>
</li>
</ol></div></div>
<h2><span class="mw-headline" id="References">References</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=60" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h2>
<h3><span class="mw-headline" id="Citations">Citations</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=61" title="Edit section: Citations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1011085734"><div class="reflist">
<div class="mw-references-wrap mw-references-columns"><ol class="references">
<li id="cite_note-norton-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-norton_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1133582631">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free a,.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:#d33}.mw-parser-output .cs1-visible-error{color:#d33}.mw-parser-output .cs1-maint{display:none;color:#3a3;margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}</style><cite id="CITEREFNortonKhokhlovUryasev2019" class="citation journal cs1">Norton, Matthew; Khokhlov, Valentyn; Uryasev, Stan (2019). <a rel="nofollow" class="external text" href="http://uryasev.ams.stonybrook.edu/wp-content/uploads/2019/10/Norton2019_CVaR_bPOE.pdf">"Calculating CVaR and bPOE for common probability distributions with application to portfolio optimization and density estimation"</a> <span class="cs1-format">(PDF)</span>. <i>Annals of Operations Research</i>. Springer. <b>299</b> (12): 12811315. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="cs1-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1811.11301">1811.11301</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10479-019-03373-1">10.1007/s10479-019-03373-1</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a>&#160;<a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:254231768">254231768</a><span class="reference-accessdate">. Retrieved <span class="nowrap">February 27,</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Annals+of+Operations+Research&amp;rft.atitle=Calculating+CVaR+and+bPOE+for+common+probability+distributions+with+application+to+portfolio+optimization+and+density+estimation&amp;rft.volume=299&amp;rft.issue=1%E2%80%932&amp;rft.pages=1281-1315&amp;rft.date=2019&amp;rft_id=info%3Aarxiv%2F1811.11301&amp;rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A254231768%23id-name%3DS2CID&amp;rft_id=info%3Adoi%2F10.1007%2Fs10479-019-03373-1&amp;rft.aulast=Norton&amp;rft.aufirst=Matthew&amp;rft.au=Khokhlov%2C+Valentyn&amp;rft.au=Uryasev%2C+Stan&amp;rft_id=http%3A%2F%2Furyasev.ams.stonybrook.edu%2Fwp-content%2Fuploads%2F2019%2F10%2FNorton2019_CVaR_bPOE.pdf&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.encyclopedia.com/topic/Normal_Distribution.aspx#3"><i>Normal Distribution</i></a>, Gale Encyclopedia of Psychology</span>
</li>
<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a href="#CITEREFCasellaBerger2001">Casella &amp; Berger (2001</a>, p.&#160;102)</span>
</li>
<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Lyon, A. (2014). <a rel="nofollow" class="external text" href="https://aidanlyon.com/normal_distributions.pdf">Why are Normal Distributions Normal?</a>, The British Journal for the Philosophy of Science.</span>
</li>
<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFJorgeStephan2006" class="citation book cs1">Jorge, Nocedal; Stephan, J. Wright (2006). <i>Numerical Optimization</i> (2nd&#160;ed.). Springer. p.&#160;249. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-0387-30303-1" title="Special:BookSources/978-0387-30303-1"><bdi>978-0387-30303-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Numerical+Optimization&amp;rft.pages=249&amp;rft.edition=2nd&amp;rft.pub=Springer&amp;rft.date=2006&amp;rft.isbn=978-0387-30303-1&amp;rft.aulast=Jorge&amp;rft.aufirst=Nocedal&amp;rft.au=Stephan%2C+J.+Wright&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></span>
</li>
<li id="cite_note-mathsisfun-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-mathsisfun_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-mathsisfun_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mathsisfun.com/data/standard-normal-distribution.html">"Normal Distribution"</a>. <i>www.mathsisfun.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">August 15,</span> 2020</span>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=unknown&amp;rft.jtitle=www.mathsisfun.com&amp;rft.atitle=Normal+Distribution&amp;rft_id=https%3A%2F%2Fwww.mathsisfun.com%2Fdata%2Fstandard-normal-distribution.html&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></span>
</li>
<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a href="#CITEREFStigler1982">Stigler (1982)</a></span>
</li>
<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><a href="#CITEREFHalperinHartleyHoel1965">Halperin, Hartley &amp; Hoel (1965</a>, item 7)</span>
</li>
<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><a href="#CITEREFMcPherson1990">McPherson (1990</a>, p.&#160;110)</span>
</li>
<li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><a href="#CITEREFBernardoSmith2000">Bernardo &amp; Smith (2000</a>, p.&#160;121)</span>
</li>
<li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFScottNowak2003" class="citation web cs1">Scott, Clayton; Nowak, Robert (August 7, 2003). <a rel="nofollow" class="external text" href="http://cnx.org/content/m11537/1.2/">"The Q-function"</a>. <i>Connexions</i>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=unknown&amp;rft.jtitle=Connexions&amp;rft.atitle=The+Q-function&amp;rft.date=2003-08-07&amp;rft.aulast=Scott&amp;rft.aufirst=Clayton&amp;rft.au=Nowak%2C+Robert&amp;rft_id=http%3A%2F%2Fcnx.org%2Fcontent%2Fm11537%2F1.2%2F&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></span>
</li>
<li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFBarak2006" class="citation web cs1">Barak, Ohad (April 6, 2006). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090325160012/http://www.eng.tau.ac.il/~jo/academic/Q.pdf">"Q Function and Error Function"</a> <span class="cs1-format">(PDF)</span>. Tel Aviv University. Archived from <a rel="nofollow" class="external text" href="http://www.eng.tau.ac.il/~jo/academic/Q.pdf">the original</a> <span class="cs1-format">(PDF)</span> on March 25, 2009.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=unknown&amp;rft.btitle=Q+Function+and+Error+Function&amp;rft.pub=Tel+Aviv+University&amp;rft.date=2006-04-06&amp;rft.aulast=Barak&amp;rft.aufirst=Ohad&amp;rft_id=http%3A%2F%2Fwww.eng.tau.ac.il%2F~jo%2Facademic%2FQ.pdf&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></span>
</li>
<li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><span class="citation mathworld" id="Reference-Mathworld-Normal_Distribution_Function"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/NormalDistributionFunction.html">"Normal Distribution Function"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=unknown&amp;rft.jtitle=MathWorld&amp;rft.atitle=Normal+Distribution+Function&amp;rft.au=Weisstein%2C+Eric+W.&amp;rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FNormalDistributionFunction.html&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></span></span>
</li>
<li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFAbramowitzStegun1983" class="citation book cs1"><a href="/wiki/Milton_Abramowitz" title="Milton Abramowitz">Abramowitz, Milton</a>; <a href="/wiki/Irene_Stegun" title="Irene Stegun">Stegun, Irene Ann</a>, eds. (1983) [June 1964]. <a rel="nofollow" class="external text" href="http://www.math.ubc.ca/~cbm/aands/page_932.htm">"Chapter 26, eqn 26.2.12"</a>. <a href="/wiki/Abramowitz_and_Stegun" title="Abramowitz and Stegun"><i>Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables</i></a>. Applied Mathematics Series. Vol.&#160;55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first&#160;ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p.&#160;932. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-0-486-61272-0" title="Special:BookSources/978-0-486-61272-0"><bdi>978-0-486-61272-0</bdi></a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a>&#160;<a rel="nofollow" class="external text" href="https://lccn.loc.gov/64-60036">64-60036</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a>&#160;<a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0167642">0167642</a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a>&#160;<a rel="nofollow" class="external text" href="https://lccn.loc.gov/65012253">65-12253</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=bookitem&amp;rft.atitle=Chapter+26%2C+eqn+26.2.12&amp;rft.btitle=Handbook+of+Mathematical+Functions+with+Formulas%2C+Graphs%2C+and+Mathematical+Tables&amp;rft.place=Washington+D.C.%3B+New+York&amp;rft.series=Applied+Mathematics+Series&amp;rft.pages=932&amp;rft.edition=Ninth+reprint+with+additional+corrections+of+tenth+original+printing+with+corrections+%28December+1972%29%3B+first&amp;rft.pub=United+States+Department+of+Commerce%2C+National+Bureau+of+Standards%3B+Dover+Publications&amp;rft.date=1983&amp;rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0167642%23id-name%3DMR&amp;rft_id=info%3Alccn%2F64-60036&amp;rft.isbn=978-0-486-61272-0&amp;rft_id=http%3A%2F%2Fwww.math.ubc.ca%2F~cbm%2Faands%2Fpage_932.htm&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></span>
</li>
<li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFCoverThomas,_Joy_A.2006" class="citation book cs1">Cover, Thomas M.; Thomas, Joy A. (2006). <span class="cs1-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/elementsinformat00cove"><i>Elements of Information Theory</i></a></span>. John Wiley and Sons. p.&#160;<a rel="nofollow" class="external text" href="https://archive.org/details/elementsinformat00cove/page/n279">254</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&#160;<a href="/wiki/Special:BookSources/9780471748816" title="Special:BookSources/9780471748816"><bdi>9780471748816</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Elements+of+Information+Theory&amp;rft.pages=254&amp;rft.pub=John+Wiley+and+Sons&amp;rft.date=2006&amp;rft.isbn=9780471748816&amp;rft.aulast=Cover&amp;rft.aufirst=Thomas+M.&amp;rft.au=Thomas%2C+Joy+A.&amp;rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Felementsinformat00cove&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></span>
</li>
<li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFParkBera2009" class="citation journal cs1">Park, Sung Y.; Bera, Anil K. (2009). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160307144515/http://wise.xmu.edu.cn/uploadfiles/paper-masterdownload/2009519932327055475115776.pdf">"Maximum Entropy Autoregressive Conditional Heteroskedasticity Model"</a> <span class="cs1-format">(PDF)</span>. <i>Journal of Econometrics</i>. <b>150</b> (2): 219230. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a>&#160;<span class="cs1-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.511.9750">10.1.1.511.9750</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jeconom.2008.12.014">10.1016/j.jeconom.2008.12.014</a>. Archived from <a rel="nofollow" class="external text" href="http://www.wise.xmu.edu.cn/Master/Download/..%5C..%5CUploadFiles%5Cpaper-masterdownload%5C2009519932327055475115776.pdf">the original</a> <span class="cs1-format">(PDF)</span> on March 7, 2016<span class="reference-accessdate">. Retrieved <span class="nowrap">June 2,</span> 2011</span>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Journal+of+Econometrics&amp;rft.atitle=Maximum+Entropy+Autoregressive+Conditional+Heteroskedasticity+Model&amp;rft.volume=150&amp;rft.issue=2&amp;rft.pages=219-230&amp;rft.date=2009&amp;rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.511.9750%23id-name%3DCiteSeerX&amp;rft_id=info%3Adoi%2F10.1016%2Fj.jeconom.2008.12.014&amp;rft.aulast=Park&amp;rft.aufirst=Sung+Y.&amp;rft.au=Bera%2C+Anil+K.&amp;rft_id=http%3A%2F%2Fwww.wise.xmu.edu.cn%2FMaster%2FDownload%2F..%255C..%255CUploadFiles%255Cpaper-masterdownload%255C2009519932327055475115776.pdf&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></span>
</li>
<li id="cite_note-Geary1936-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-Geary1936_17-0">^</a></b></span> <span class="reference-text">Geary RC(1936) The distribution of the "Student's ratio for the non-normal samples". Supplement to the Journal of the Royal Statistical Society 3 (2): 178184</span>
</li>
<li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFLukacs,_Eugene1942" class="citation journal cs1"><a href="/wiki/Eugene_Lukacs" title="Eugene Lukacs">Lukacs, Eugene</a> (March 1942). "A Characterization of the Normal Distribution". <i><a href="/wiki/Annals_of_Mathematical_Statistics" title="Annals of Mathematical Statistics">Annals of Mathematical Statistics</a></i>. <b>13</b> (1): 9193. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1214%2FAOMS%2F1177731647">10.1214/AOMS/1177731647</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a>&#160;<a rel="nofollow" class="external text" href="https://www.worldcat.org/issn/0003-4851">0003-4851</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a>&#160;<a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2236166">2236166</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a>&#160;<a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0006626">0006626</a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a>&#160;<a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&amp;q=an:0060.28509">0060.28509</a>. <a href="/wiki/WDQ_(identifier)" class="mw-redirect" title="WDQ (identifier)">Wikidata</a>&#160;<a href="https://www.wikidata.org/wiki/Q55897617" class="extiw" title="d:Q55897617">Q55897617</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Annals+of+Mathematical+Statistics&amp;rft.atitle=A+Characterization+of+the+Normal+Distribution&amp;rft.volume=13&amp;rft.issue=1&amp;rft.pages=91-93&amp;rft.date=1942-03&amp;rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0060.28509%23id-name%3DZbl&amp;rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2236166%23id-name%3DJSTOR&amp;rft_id=info%3Adoi%2F10.1214%2FAOMS%2F1177731647&amp;rft.issn=0003-4851&amp;rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D6626%23id-name%3DMR&amp;rft.au=Lukacs%2C+Eugene&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></span>
</li>
<li id="cite_note-PR2.1.4-19"><span class="mw-cite-backlink">^ <a href="#cite_ref-PR2.1.4_19-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-PR2.1.4_19-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-PR2.1.4_19-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFPatelRead1996">Patel &amp; Read (1996</a>, [2.1.4])</span>
</li>
<li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><a href="#CITEREFFan1991">Fan (1991</a>, p.&#160;1258)</span>
</li>
<li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><a href="#CITEREFPatelRead1996">Patel &amp; Read (1996</a>, [2.1.8])</span>
</li>
<li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFPapoulis" class="citation book cs1">Papoulis, Athanasios. <i>Probability, Random Variables and Stochastic Processes</i> (4th&#160;ed.). p.&#160;148.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Probability%2C+Random+Variables+and+Stochastic+Processes&amp;rft.pages=148&amp;rft.edition=4th&amp;rft.aulast=Papoulis&amp;rft.aufirst=Athanasios&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></span>
</li>
<li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><a href="#CITEREFBryc1995">Bryc (1995</a>, p.&#160;23)</span>
</li>
<li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><a href="#CITEREFBryc1995">Bryc (1995</a>, p.&#160;24)</span>
</li>
<li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><a href="#CITEREFCoverThomas2006">Cover &amp; Thomas (2006</a>, p.&#160;254)</span>
</li>
<li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFWilliams2001" class="citation book cs1">Williams, David (2001). <span class="cs1-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/weighingoddscour00will"><i>Weighing the odds&#160;: a course in probability and statistics</i></a></span> (Reprinted.&#160;ed.). Cambridge [u.a.]: Cambridge Univ. Press. pp.&#160;<a rel="nofollow" class="external text" href="https://archive.org/details/weighingoddscour00will/page/n219">197</a>199. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-0-521-00618-7" title="Special:BookSources/978-0-521-00618-7"><bdi>978-0-521-00618-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Weighing+the+odds+%3A+a+course+in+probability+and+statistics&amp;rft.place=Cambridge+%5Bu.a.%5D&amp;rft.pages=197-199&amp;rft.edition=Reprinted.&amp;rft.pub=Cambridge+Univ.+Press&amp;rft.date=2001&amp;rft.isbn=978-0-521-00618-7&amp;rft.aulast=Williams&amp;rft.aufirst=David&amp;rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fweighingoddscour00will&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></span>
</li>
<li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFSmith2000" class="citation book cs1">Smith, José M. Bernardo; Adrian F. M. (2000). <span class="cs1-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/bayesiantheory00bern_963"><i>Bayesian theory</i></a></span> (Reprint&#160;ed.). Chichester [u.a.]: Wiley. pp.&#160;<a rel="nofollow" class="external text" href="https://archive.org/details/bayesiantheory00bern_963/page/n224">209</a>, 366. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-0-471-49464-5" title="Special:BookSources/978-0-471-49464-5"><bdi>978-0-471-49464-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Bayesian+theory&amp;rft.place=Chichester+%5Bu.a.%5D&amp;rft.pages=209%2C+366&amp;rft.edition=Reprint&amp;rft.pub=Wiley&amp;rft.date=2000&amp;rft.isbn=978-0-471-49464-5&amp;rft.aulast=Smith&amp;rft.aufirst=Jos%C3%A9+M.+Bernardo%3B+Adrian+F.+M.&amp;rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fbayesiantheory00bern_963&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>)</span></span>
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<li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text">O'Hagan, A. (1994) <i>Kendall's Advanced Theory of statistics, Vol 2B, Bayesian Inference</i>, Edward Arnold. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&#160;<a href="/wiki/Special:BookSources/0-340-52922-9" title="Special:BookSources/0-340-52922-9">0-340-52922-9</a> (Section 5.40)</span>
</li>
<li id="cite_note-Bryc_1995_35-29"><span class="mw-cite-backlink">^ <a href="#cite_ref-Bryc_1995_35_29-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Bryc_1995_35_29-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFBryc1995">Bryc (1995</a>, p.&#160;35)</span>
</li>
<li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.math.uiuc.edu/~r-ash/Stat/StatLec21-25.pdf">UIUC, Lecture 21. <i>The Multivariate Normal Distribution</i></a>, 21.6:"Individually Gaussian Versus Jointly Gaussian".</span>
</li>
<li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text">Edward L. Melnick and Aaron Tenenbein, "Misspecifications of the Normal Distribution", <i><a href="/wiki/The_American_Statistician" title="The American Statistician">The American Statistician</a></i>, volume 36, number 4 November 1982, pages 372373</span>
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<li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.allisons.org/ll/MML/KL/Normal/">"Kullback Leibler (KL) Distance of Two Normal (Gaussian) Probability Distributions"</a>. <i>Allisons.org</i>. December 5, 2007<span class="reference-accessdate">. Retrieved <span class="nowrap">March 3,</span> 2017</span>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=unknown&amp;rft.jtitle=Allisons.org&amp;rft.atitle=Kullback+Leibler+%28KL%29+Distance+of+Two+Normal+%28Gaussian%29+Probability+Distributions&amp;rft.date=2007-12-05&amp;rft_id=http%3A%2F%2Fwww.allisons.org%2Fll%2FMML%2FKL%2FNormal%2F&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></span>
</li>
<li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFJordan2010" class="citation web cs1">Jordan, Michael I. (February 8, 2010). <a rel="nofollow" class="external text" href="http://www.cs.berkeley.edu/~jordan/courses/260-spring10/lectures/lecture5.pdf">"Stat260: Bayesian Modeling and Inference: The Conjugate Prior for the Normal Distribution"</a> <span class="cs1-format">(PDF)</span>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=unknown&amp;rft.btitle=Stat260%3A+Bayesian+Modeling+and+Inference%3A+The+Conjugate+Prior+for+the+Normal+Distribution&amp;rft.date=2010-02-08&amp;rft.aulast=Jordan&amp;rft.aufirst=Michael+I.&amp;rft_id=http%3A%2F%2Fwww.cs.berkeley.edu%2F~jordan%2Fcourses%2F260-spring10%2Flectures%2Flecture5.pdf&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></span>
</li>
<li id="cite_note-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-34">^</a></b></span> <span class="reference-text"><a href="#CITEREFAmariNagaoka2000">Amari &amp; Nagaoka (2000)</a></span>
</li>
<li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.stat.ucla.edu/~dinov/courses_students.dir/Applets.dir/NormalApprox2PoissonApplet.html">"Normal Approximation to Poisson Distribution"</a>. <i>Stat.ucla.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">March 3,</span> 2017</span>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=unknown&amp;rft.jtitle=Stat.ucla.edu&amp;rft.atitle=Normal+Approximation+to+Poisson+Distribution&amp;rft_id=http%3A%2F%2Fwww.stat.ucla.edu%2F~dinov%2Fcourses_students.dir%2FApplets.dir%2FNormalApprox2PoissonApplet.html&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></span>
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<li id="cite_note-Das-36"><span class="mw-cite-backlink">^ <a href="#cite_ref-Das_36-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Das_36-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFDas2021" class="citation journal cs1">Das, Abhranil (2021). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8419883">"A method to integrate and classify normal distributions"</a>. <i>Journal of Vision</i>. <b>21</b> (10): 1. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="cs1-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2012.14331">2012.14331</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1167%2Fjov.21.10.1">10.1167/jov.21.10.1</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a>&#160;<span class="cs1-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8419883">8419883</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a>&#160;<a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/34468706">34468706</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Journal+of+Vision&amp;rft.atitle=A+method+to+integrate+and+classify+normal+distributions&amp;rft.volume=21&amp;rft.issue=10&amp;rft.pages=1&amp;rft.date=2021&amp;rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC8419883%23id-name%3DPMC&amp;rft_id=info%3Apmid%2F34468706&amp;rft_id=info%3Aarxiv%2F2012.14331&amp;rft_id=info%3Adoi%2F10.1167%2Fjov.21.10.1&amp;rft.aulast=Das&amp;rft.aufirst=Abhranil&amp;rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC8419883&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></span>
</li>
<li id="cite_note-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-37">^</a></b></span> <span class="reference-text"><a href="#CITEREFBryc1995">Bryc (1995</a>, p.&#160;27)</span>
</li>
<li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/NormalProductDistribution.html">"Normal Product Distribution"</a>. <i>MathWorld</i>. wolfram.com.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=unknown&amp;rft.jtitle=MathWorld&amp;rft.atitle=Normal+Product+Distribution&amp;rft.aulast=Weisstein&amp;rft.aufirst=Eric+W.&amp;rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FNormalProductDistribution.html&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></span>
</li>
<li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFLukacs1942" class="citation journal cs1">Lukacs, Eugene (1942). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faoms%2F1177731647">"A Characterization of the Normal Distribution"</a>. <i><a href="/wiki/The_Annals_of_Mathematical_Statistics" class="mw-redirect" title="The Annals of Mathematical Statistics">The Annals of Mathematical Statistics</a></i>. <b>13</b> (1): 913. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="cs1-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faoms%2F1177731647">10.1214/aoms/1177731647</a></span>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a>&#160;<a rel="nofollow" class="external text" href="https://www.worldcat.org/issn/0003-4851">0003-4851</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a>&#160;<a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2236166">2236166</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=The+Annals+of+Mathematical+Statistics&amp;rft.atitle=A+Characterization+of+the+Normal+Distribution&amp;rft.volume=13&amp;rft.issue=1&amp;rft.pages=91-3&amp;rft.date=1942&amp;rft.issn=0003-4851&amp;rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2236166%23id-name%3DJSTOR&amp;rft_id=info%3Adoi%2F10.1214%2Faoms%2F1177731647&amp;rft.aulast=Lukacs&amp;rft.aufirst=Eugene&amp;rft_id=https%3A%2F%2Fdoi.org%2F10.1214%252Faoms%252F1177731647&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></span>
</li>
<li id="cite_note-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-40">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFBasuLaha1954" class="citation journal cs1">Basu, D.; Laha, R. G. (1954). "On Some Characterizations of the Normal Distribution". <i><a href="/wiki/Sankhy%C4%81_(journal)" class="mw-redirect" title="Sankhyā (journal)">Sankhyā</a></i>. <b>13</b> (4): 35962. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a>&#160;<a rel="nofollow" class="external text" href="https://www.worldcat.org/issn/0036-4452">0036-4452</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a>&#160;<a rel="nofollow" class="external text" href="https://www.jstor.org/stable/25048183">25048183</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Sankhy%C4%81&amp;rft.atitle=On+Some+Characterizations+of+the+Normal+Distribution&amp;rft.volume=13&amp;rft.issue=4&amp;rft.pages=359-62&amp;rft.date=1954&amp;rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F25048183%23id-name%3DJSTOR&amp;rft.issn=0036-4452&amp;rft.aulast=Basu&amp;rft.aufirst=D.&amp;rft.au=Laha%2C+R.+G.&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></span>
</li>
<li id="cite_note-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-41">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFLehmann1997" class="citation book cs1">Lehmann, E. L. (1997). <i>Testing Statistical Hypotheses</i> (2nd&#160;ed.). Springer. p.&#160;199. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-0-387-94919-2" title="Special:BookSources/978-0-387-94919-2"><bdi>978-0-387-94919-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Testing+Statistical+Hypotheses&amp;rft.pages=199&amp;rft.edition=2nd&amp;rft.pub=Springer&amp;rft.date=1997&amp;rft.isbn=978-0-387-94919-2&amp;rft.aulast=Lehmann&amp;rft.aufirst=E.+L.&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></span>
</li>
<li id="cite_note-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-42">^</a></b></span> <span class="reference-text"><a href="#CITEREFPatelRead1996">Patel &amp; Read (1996</a>, [2.3.6])</span>
</li>
<li id="cite_note-43"><span class="mw-cite-backlink"><b><a href="#cite_ref-43">^</a></b></span> <span class="reference-text"><a href="#CITEREFGalambosSimonelli2004">Galambos &amp; Simonelli (2004</a>, Theorem&#160;3.5)</span>
</li>
<li id="cite_note-LK-44"><span class="mw-cite-backlink">^ <a href="#cite_ref-LK_44-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-LK_44-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFLukacsKing1954">Lukacs &amp; King (1954)</a></span>
</li>
<li id="cite_note-45"><span class="mw-cite-backlink"><b><a href="#cite_ref-45">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFQuine1993" class="citation journal cs1">Quine, M.P. (1993). <a rel="nofollow" class="external text" href="http://www.math.uni.wroc.pl/~pms/publicationsArticle.php?nr=14.2&amp;nrA=8&amp;ppB=257&amp;ppE=263">"On three characterisations of the normal distribution"</a>. <i>Probability and Mathematical Statistics</i>. <b>14</b> (2): 257263.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Probability+and+Mathematical+Statistics&amp;rft.atitle=On+three+characterisations+of+the+normal+distribution&amp;rft.volume=14&amp;rft.issue=2&amp;rft.pages=257-263&amp;rft.date=1993&amp;rft.aulast=Quine&amp;rft.aufirst=M.P.&amp;rft_id=http%3A%2F%2Fwww.math.uni.wroc.pl%2F~pms%2FpublicationsArticle.php%3Fnr%3D14.2%26nrA%3D8%26ppB%3D257%26ppE%3D263&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></span>
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<li id="cite_note-John1982-46"><span class="mw-cite-backlink"><b><a href="#cite_ref-John1982_46-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFJohn1982" class="citation journal cs1">John, S (1982). "The three parameter two-piece normal family of distributions and its fitting". <i>Communications in Statistics - Theory and Methods</i>. <b>11</b> (8): 879885. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F03610928208828279">10.1080/03610928208828279</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Communications+in+Statistics+-+Theory+and+Methods&amp;rft.atitle=The+three+parameter+two-piece+normal+family+of+distributions+and+its+fitting&amp;rft.volume=11&amp;rft.issue=8&amp;rft.pages=879-885&amp;rft.date=1982&amp;rft_id=info%3Adoi%2F10.1080%2F03610928208828279&amp;rft.aulast=John&amp;rft.aufirst=S&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></span>
</li>
<li id="cite_note-Kri127-47"><span class="mw-cite-backlink">^ <a href="#cite_ref-Kri127_47-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Kri127_47-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFKrishnamoorthy2006">Krishnamoorthy (2006</a>, p.&#160;127)</span>
</li>
<li id="cite_note-48"><span class="mw-cite-backlink"><b><a href="#cite_ref-48">^</a></b></span> <span class="reference-text"><a href="#CITEREFKrishnamoorthy2006">Krishnamoorthy (2006</a>, p.&#160;130)</span>
</li>
<li id="cite_note-49"><span class="mw-cite-backlink"><b><a href="#cite_ref-49">^</a></b></span> <span class="reference-text"><a href="#CITEREFKrishnamoorthy2006">Krishnamoorthy (2006</a>, p.&#160;133)</span>
</li>
<li id="cite_note-50"><span class="mw-cite-backlink"><b><a href="#cite_ref-50">^</a></b></span> <span class="reference-text"><a href="#CITEREFHuxley1932">Huxley (1932)</a></span>
</li>
<li id="cite_note-51"><span class="mw-cite-backlink"><b><a href="#cite_ref-51">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFJaynes2003" class="citation book cs1">Jaynes, Edwin T. (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=tTN4HuUNXjgC&amp;pg=PA592"><i>Probability Theory: The Logic of Science</i></a>. Cambridge University Press. pp.&#160;592593. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&#160;<a href="/wiki/Special:BookSources/9780521592710" title="Special:BookSources/9780521592710"><bdi>9780521592710</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Probability+Theory%3A+The+Logic+of+Science&amp;rft.pages=592-593&amp;rft.pub=Cambridge+University+Press&amp;rft.date=2003&amp;rft.isbn=9780521592710&amp;rft.aulast=Jaynes&amp;rft.aufirst=Edwin+T.&amp;rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DtTN4HuUNXjgC%26pg%3DPA592&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></span>
</li>
<li id="cite_note-52"><span class="mw-cite-backlink"><b><a href="#cite_ref-52">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFOosterbaan1994" class="citation book cs1">Oosterbaan, Roland J. (1994). <a rel="nofollow" class="external text" href="http://www.waterlog.info/pdf/freqtxt.pdf">"Chapter 6: Frequency and Regression Analysis of Hydrologic Data"</a> <span class="cs1-format">(PDF)</span>. In Ritzema, Henk P. (ed.). <i>Drainage Principles and Applications, Publication 16</i> (second revised&#160;ed.). Wageningen, The Netherlands: International Institute for Land Reclamation and Improvement (ILRI). pp.&#160;175224. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-90-70754-33-4" title="Special:BookSources/978-90-70754-33-4"><bdi>978-90-70754-33-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=bookitem&amp;rft.atitle=Chapter+6%3A+Frequency+and+Regression+Analysis+of+Hydrologic+Data&amp;rft.btitle=Drainage+Principles+and+Applications%2C+Publication+16&amp;rft.place=Wageningen%2C+The+Netherlands&amp;rft.pages=175-224&amp;rft.edition=second+revised&amp;rft.pub=International+Institute+for+Land+Reclamation+and+Improvement+%28ILRI%29&amp;rft.date=1994&amp;rft.isbn=978-90-70754-33-4&amp;rft.aulast=Oosterbaan&amp;rft.aufirst=Roland+J.&amp;rft_id=http%3A%2F%2Fwww.waterlog.info%2Fpdf%2Ffreqtxt.pdf&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></span>
</li>
<li id="cite_note-53"><span class="mw-cite-backlink"><b><a href="#cite_ref-53">^</a></b></span> <span class="reference-text">Why Most Published Research Findings Are False, John P. A. Ioannidis, 2005</span>
</li>
<li id="cite_note-54"><span class="mw-cite-backlink"><b><a href="#cite_ref-54">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFWichura1988" class="citation journal cs1">Wichura, Michael J. (1988). "Algorithm AS241: The Percentage Points of the Normal Distribution". <i>Applied Statistics</i>. <b>37</b> (3): 47784. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2347330">10.2307/2347330</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a>&#160;<a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2347330">2347330</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Applied+Statistics&amp;rft.atitle=Algorithm+AS241%3A+The+Percentage+Points+of+the+Normal+Distribution&amp;rft.volume=37&amp;rft.issue=3&amp;rft.pages=477-84&amp;rft.date=1988&amp;rft_id=info%3Adoi%2F10.2307%2F2347330&amp;rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2347330%23id-name%3DJSTOR&amp;rft.aulast=Wichura&amp;rft.aufirst=Michael+J.&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></span>
</li>
<li id="cite_note-55"><span class="mw-cite-backlink"><b><a href="#cite_ref-55">^</a></b></span> <span class="reference-text"><a href="#CITEREFJohnsonKotzBalakrishnan1995">Johnson, Kotz &amp; Balakrishnan (1995</a>, Equation (26.48))</span>
</li>
<li id="cite_note-56"><span class="mw-cite-backlink"><b><a href="#cite_ref-56">^</a></b></span> <span class="reference-text"><a href="#CITEREFKindermanMonahan1977">Kinderman &amp; Monahan (1977)</a></span>
</li>
<li id="cite_note-57"><span class="mw-cite-backlink"><b><a href="#cite_ref-57">^</a></b></span> <span class="reference-text"><a href="#CITEREFLeva1992">Leva (1992)</a></span>
</li>
<li id="cite_note-58"><span class="mw-cite-backlink"><b><a href="#cite_ref-58">^</a></b></span> <span class="reference-text"><a href="#CITEREFMarsagliaTsang2000">Marsaglia &amp; Tsang (2000)</a></span>
</li>
<li id="cite_note-59"><span class="mw-cite-backlink"><b><a href="#cite_ref-59">^</a></b></span> <span class="reference-text"><a href="#CITEREFKarney2016">Karney (2016)</a></span>
</li>
<li id="cite_note-60"><span class="mw-cite-backlink"><b><a href="#cite_ref-60">^</a></b></span> <span class="reference-text"><a href="#CITEREFMonahan1985">Monahan (1985</a>, section 2)</span>
</li>
<li id="cite_note-61"><span class="mw-cite-backlink"><b><a href="#cite_ref-61">^</a></b></span> <span class="reference-text"><a href="#CITEREFWallace1996">Wallace (1996)</a></span>
</li>
<li id="cite_note-63"><span class="mw-cite-backlink"><b><a href="#cite_ref-63">^</a></b></span> <span class="reference-text"><a href="#CITEREFJohnsonKotzBalakrishnan1994">Johnson, Kotz &amp; Balakrishnan (1994</a>, p.&#160;85)</span>
</li>
<li id="cite_note-64"><span class="mw-cite-backlink"><b><a href="#cite_ref-64">^</a></b></span> <span class="reference-text"><a href="#CITEREFLe_CamLo_Yang2000">Le Cam &amp; Lo Yang (2000</a>, p.&#160;74)</span>
</li>
<li id="cite_note-66"><span class="mw-cite-backlink"><b><a href="#cite_ref-66">^</a></b></span> <span class="reference-text">De Moivre, Abraham (1733), Corollary I see <a href="#CITEREFWalker1985">Walker (1985</a>, p.&#160;77)</span>
</li>
<li id="cite_note-67"><span class="mw-cite-backlink"><b><a href="#cite_ref-67">^</a></b></span> <span class="reference-text"><a href="#CITEREFStigler1986">Stigler (1986</a>, p.&#160;76)</span>
</li>
<li id="cite_note-69"><span class="mw-cite-backlink"><b><a href="#cite_ref-69">^</a></b></span> <span class="reference-text"><a href="#CITEREFGauss1809">Gauss (1809</a>, section 177)</span>
</li>
<li id="cite_note-70"><span class="mw-cite-backlink"><b><a href="#cite_ref-70">^</a></b></span> <span class="reference-text"><a href="#CITEREFGauss1809">Gauss (1809</a>, section 179)</span>
</li>
<li id="cite_note-72"><span class="mw-cite-backlink"><b><a href="#cite_ref-72">^</a></b></span> <span class="reference-text"><a href="#CITEREFLaplace1774">Laplace (1774</a>, Problem III)</span>
</li>
<li id="cite_note-73"><span class="mw-cite-backlink"><b><a href="#cite_ref-73">^</a></b></span> <span class="reference-text"><a href="#CITEREFPearson1905">Pearson (1905</a>, p.&#160;189)</span>
</li>
<li id="cite_note-74"><span class="mw-cite-backlink"><b><a href="#cite_ref-74">^</a></b></span> <span class="reference-text"><a href="#CITEREFStigler1986">Stigler (1986</a>, p.&#160;144)</span>
</li>
<li id="cite_note-75"><span class="mw-cite-backlink"><b><a href="#cite_ref-75">^</a></b></span> <span class="reference-text"><a href="#CITEREFStigler1978">Stigler (1978</a>, p.&#160;243)</span>
</li>
<li id="cite_note-76"><span class="mw-cite-backlink"><b><a href="#cite_ref-76">^</a></b></span> <span class="reference-text"><a href="#CITEREFStigler1978">Stigler (1978</a>, p.&#160;244)</span>
</li>
<li id="cite_note-77"><span class="mw-cite-backlink"><b><a href="#cite_ref-77">^</a></b></span> <span class="reference-text"><a href="#CITEREFMaxwell1860">Maxwell (1860</a>, p.&#160;23)</span>
</li>
<li id="cite_note-78"><span class="mw-cite-backlink"><b><a href="#cite_ref-78">^</a></b></span> <span class="reference-text">Jaynes, Edwin J.; <i>Probability Theory: The Logic of Science</i>, <a rel="nofollow" class="external text" href="http://www-biba.inrialpes.fr/Jaynes/cc07s.pdf">Ch. 7</a>.</span>
</li>
<li id="cite_note-80"><span class="mw-cite-backlink"><b><a href="#cite_ref-80">^</a></b></span> <span class="reference-text">Peirce, Charles S. (c. 1909 MS), <i><a href="/wiki/Charles_Sanders_Peirce_bibliography#CP" title="Charles Sanders Peirce bibliography">Collected Papers</a></i> v. 6, paragraph 327.</span>
</li>
<li id="cite_note-81"><span class="mw-cite-backlink"><b><a href="#cite_ref-81">^</a></b></span> <span class="reference-text"><a href="#CITEREFKruskalStigler1997">Kruskal &amp; Stigler (1997)</a>.</span>
</li>
<li id="cite_note-82"><span class="mw-cite-backlink"><b><a href="#cite_ref-82">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://jeff560.tripod.com/s.html">"Earliest Uses... (Entry Standard Normal Curve)"</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=unknown&amp;rft.btitle=Earliest+Uses...+%28Entry+Standard+Normal+Curve%29&amp;rft_id=http%3A%2F%2Fjeff560.tripod.com%2Fs.html&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></span>
</li>
<li id="cite_note-Sun,_Kong_and_Pal-83"><span class="mw-cite-backlink"><b><a href="#cite_ref-Sun,_Kong_and_Pal_83-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFSunKongPal2021" class="citation journal cs1">Sun, Jingchao; Kong, Maiying; Pal, Subhadip (June 22, 2021). <a rel="nofollow" class="external text" href="https://www.tandfonline.com/doi/abs/10.1080/03610926.2021.1934700?journalCode=lsta20">"The Modified-Half-Normal distribution: Properties and an efficient sampling scheme"</a>. <i>Communications in Statistics - Theory and Methods</i>. <b>52</b> (5): 15911613. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F03610926.2021.1934700">10.1080/03610926.2021.1934700</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a>&#160;<a rel="nofollow" class="external text" href="https://www.worldcat.org/issn/0361-0926">0361-0926</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a>&#160;<a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:237919587">237919587</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Communications+in+Statistics+-+Theory+and+Methods&amp;rft.atitle=The+Modified-Half-Normal+distribution%3A+Properties+and+an+efficient+sampling+scheme&amp;rft.volume=52&amp;rft.issue=5&amp;rft.pages=1591-1613&amp;rft.date=2021-06-22&amp;rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A237919587%23id-name%3DS2CID&amp;rft.issn=0361-0926&amp;rft_id=info%3Adoi%2F10.1080%2F03610926.2021.1934700&amp;rft.aulast=Sun&amp;rft.aufirst=Jingchao&amp;rft.au=Kong%2C+Maiying&amp;rft.au=Pal%2C+Subhadip&amp;rft_id=https%3A%2F%2Fwww.tandfonline.com%2Fdoi%2Fabs%2F10.1080%2F03610926.2021.1934700%3FjournalCode%3Dlsta20&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></span>
</li>
</ol></div></div>
<h3><span class="mw-headline" id="Sources">Sources</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=62" title="Edit section: Sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h3>
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<ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFAldrichMiller" class="citation web cs1">Aldrich, John; Miller, Jeff. <a rel="nofollow" class="external text" href="http://jeff560.tripod.com/stat.html">"Earliest Uses of Symbols in Probability and Statistics"</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=unknown&amp;rft.btitle=Earliest+Uses+of+Symbols+in+Probability+and+Statistics&amp;rft.aulast=Aldrich&amp;rft.aufirst=John&amp;rft.au=Miller%2C+Jeff&amp;rft_id=http%3A%2F%2Fjeff560.tripod.com%2Fstat.html&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFAldrichMiller" class="citation web cs1">Aldrich, John; Miller, Jeff. <a rel="nofollow" class="external text" href="http://jeff560.tripod.com/mathword.html">"Earliest Known Uses of Some of the Words of Mathematics"</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=unknown&amp;rft.btitle=Earliest+Known+Uses+of+Some+of+the+Words+of+Mathematics&amp;rft.aulast=Aldrich&amp;rft.aufirst=John&amp;rft.au=Miller%2C+Jeff&amp;rft_id=http%3A%2F%2Fjeff560.tripod.com%2Fmathword.html&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span> In particular, the entries for <a rel="nofollow" class="external text" href="http://jeff560.tripod.com/b.html">"bell-shaped and bell curve"</a>, <a rel="nofollow" class="external text" href="http://jeff560.tripod.com/n.html">"normal (distribution)"</a>, <a rel="nofollow" class="external text" href="http://jeff560.tripod.com/g.html">"Gaussian"</a>, and <a rel="nofollow" class="external text" href="http://jeff560.tripod.com/e.html">"Error, law of error, theory of errors, etc."</a>.</li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFAmariNagaoka2000" class="citation book cs1">Amari, Shun-ichi; Nagaoka, Hiroshi (2000). <i>Methods of Information Geometry</i>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-0-8218-0531-2" title="Special:BookSources/978-0-8218-0531-2"><bdi>978-0-8218-0531-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Methods+of+Information+Geometry&amp;rft.pub=Oxford+University+Press&amp;rft.date=2000&amp;rft.isbn=978-0-8218-0531-2&amp;rft.aulast=Amari&amp;rft.aufirst=Shun-ichi&amp;rft.au=Nagaoka%2C+Hiroshi&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFBernardoSmith2000" class="citation book cs1">Bernardo, José M.; Smith, Adrian F. M. (2000). <i>Bayesian Theory</i>. Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-0-471-49464-5" title="Special:BookSources/978-0-471-49464-5"><bdi>978-0-471-49464-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Bayesian+Theory&amp;rft.pub=Wiley&amp;rft.date=2000&amp;rft.isbn=978-0-471-49464-5&amp;rft.aulast=Bernardo&amp;rft.aufirst=Jos%C3%A9+M.&amp;rft.au=Smith%2C+Adrian+F.+M.&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFBryc1995" class="citation book cs1">Bryc, Wlodzimierz (1995). <i>The Normal Distribution: Characterizations with Applications</i>. Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-0-387-97990-8" title="Special:BookSources/978-0-387-97990-8"><bdi>978-0-387-97990-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=The+Normal+Distribution%3A+Characterizations+with+Applications&amp;rft.pub=Springer-Verlag&amp;rft.date=1995&amp;rft.isbn=978-0-387-97990-8&amp;rft.aulast=Bryc&amp;rft.aufirst=Wlodzimierz&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFCasellaBerger2001" class="citation book cs1">Casella, George; Berger, Roger L. (2001). <i>Statistical Inference</i> (2nd&#160;ed.). Duxbury. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-0-534-24312-8" title="Special:BookSources/978-0-534-24312-8"><bdi>978-0-534-24312-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Statistical+Inference&amp;rft.edition=2nd&amp;rft.pub=Duxbury&amp;rft.date=2001&amp;rft.isbn=978-0-534-24312-8&amp;rft.aulast=Casella&amp;rft.aufirst=George&amp;rft.au=Berger%2C+Roger+L.&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFCody1969" class="citation journal cs1">Cody, William J. (1969). <a href="/wiki/Error_function#cite_note-5" title="Error function">"Rational Chebyshev Approximations for the Error Function"</a>. <i>Mathematics of Computation</i>. <b>23</b> (107): 631638. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="cs1-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0025-5718-1969-0247736-4">10.1090/S0025-5718-1969-0247736-4</a></span>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Mathematics+of+Computation&amp;rft.atitle=Rational+Chebyshev+Approximations+for+the+Error+Function&amp;rft.volume=23&amp;rft.issue=107&amp;rft.pages=631-638&amp;rft.date=1969&amp;rft_id=info%3Adoi%2F10.1090%2FS0025-5718-1969-0247736-4&amp;rft.aulast=Cody&amp;rft.aufirst=William+J.&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFCoverThomas2006" class="citation book cs1">Cover, Thomas M.; Thomas, Joy A. (2006). <i>Elements of Information Theory</i>. John Wiley and Sons.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Elements+of+Information+Theory&amp;rft.pub=John+Wiley+and+Sons&amp;rft.date=2006&amp;rft.aulast=Cover&amp;rft.aufirst=Thomas+M.&amp;rft.au=Thomas%2C+Joy+A.&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFDia2023" class="citation journal cs1">Dia, Yaya D. (2023). <a rel="nofollow" class="external text" href="https://ssrn.com/abstract=4487559">"Approximate Incomplete Integrals, Application to Complementary Error Function"</a>. <i>SSRN</i>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2139%2Fssrn.4487559">10.2139/ssrn.4487559</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a>&#160;<a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:259689086">259689086</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=SSRN&amp;rft.atitle=Approximate+Incomplete+Integrals%2C+Application+to+Complementary+Error+Function&amp;rft.date=2023&amp;rft_id=info%3Adoi%2F10.2139%2Fssrn.4487559&amp;rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A259689086%23id-name%3DS2CID&amp;rft.aulast=Dia&amp;rft.aufirst=Yaya+D.&amp;rft_id=https%3A%2F%2Fssrn.com%2Fabstract%3D4487559&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFde_Moivre1738" class="citation book cs1"><a href="/wiki/Abraham_de_Moivre" title="Abraham de Moivre">de Moivre, Abraham</a> (1738). <a href="/wiki/The_Doctrine_of_Chances" title="The Doctrine of Chances"><i>The Doctrine of Chances</i></a>. American Mathematical Society. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-0-8218-2103-9" title="Special:BookSources/978-0-8218-2103-9"><bdi>978-0-8218-2103-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=The+Doctrine+of+Chances&amp;rft.pub=American+Mathematical+Society&amp;rft.date=1738&amp;rft.isbn=978-0-8218-2103-9&amp;rft.aulast=de+Moivre&amp;rft.aufirst=Abraham&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFFan1991" class="citation journal cs1">Fan, Jianqing (1991). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176348248">"On the optimal rates of convergence for nonparametric deconvolution problems"</a>. <i>The Annals of Statistics</i>. <b>19</b> (3): 12571272. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="cs1-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176348248">10.1214/aos/1176348248</a></span>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a>&#160;<a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2241949">2241949</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=The+Annals+of+Statistics&amp;rft.atitle=On+the+optimal+rates+of+convergence+for+nonparametric+deconvolution+problems&amp;rft.volume=19&amp;rft.issue=3&amp;rft.pages=1257-1272&amp;rft.date=1991&amp;rft_id=info%3Adoi%2F10.1214%2Faos%2F1176348248&amp;rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2241949%23id-name%3DJSTOR&amp;rft.aulast=Fan&amp;rft.aufirst=Jianqing&amp;rft_id=https%3A%2F%2Fdoi.org%2F10.1214%252Faos%252F1176348248&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFGalton1889" class="citation book cs1">Galton, Francis (1889). <a rel="nofollow" class="external text" href="http://galton.org/books/natural-inheritance/pdf/galton-nat-inh-1up-clean.pdf"><i>Natural Inheritance</i></a> <span class="cs1-format">(PDF)</span>. London, UK: Richard Clay and Sons.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Natural+Inheritance&amp;rft.place=London%2C+UK&amp;rft.pub=Richard+Clay+and+Sons&amp;rft.date=1889&amp;rft.aulast=Galton&amp;rft.aufirst=Francis&amp;rft_id=http%3A%2F%2Fgalton.org%2Fbooks%2Fnatural-inheritance%2Fpdf%2Fgalton-nat-inh-1up-clean.pdf&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFGalambosSimonelli2004" class="citation book cs1">Galambos, Janos; Simonelli, Italo (2004). <span class="cs1-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/productsofrandom00gala"><i>Products of Random Variables: Applications to Problems of Physics and to Arithmetical Functions</i></a></span>. Marcel Dekker, Inc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-0-8247-5402-0" title="Special:BookSources/978-0-8247-5402-0"><bdi>978-0-8247-5402-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Products+of+Random+Variables%3A+Applications+to+Problems+of+Physics+and+to+Arithmetical+Functions&amp;rft.pub=Marcel+Dekker%2C+Inc.&amp;rft.date=2004&amp;rft.isbn=978-0-8247-5402-0&amp;rft.aulast=Galambos&amp;rft.aufirst=Janos&amp;rft.au=Simonelli%2C+Italo&amp;rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fproductsofrandom00gala&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFGauss1809" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss, Carolo Friderico</a> (1809). <a rel="nofollow" class="external text" href="https://archive.org/details/theoriamotuscor00gausgoog"><i>Theoria motvs corporvm coelestivm in sectionibvs conicis Solem ambientivm</i></a> &#91;<i>Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections</i>&#93; (in Latin). Hambvrgi, Svmtibvs F. Perthes et I. H. Besser. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1TIAAAAAQAAJ">English translation</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Theoria+motvs+corporvm+coelestivm+in+sectionibvs+conicis+Solem+ambientivm&amp;rft.pub=Hambvrgi%2C+Svmtibvs+F.+Perthes+et+I.+H.+Besser&amp;rft.date=1809&amp;rft.aulast=Gauss&amp;rft.aufirst=Carolo+Friderico&amp;rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ftheoriamotuscor00gausgoog&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFGould1981" class="citation book cs1"><a href="/wiki/Stephen_Jay_Gould" title="Stephen Jay Gould">Gould, Stephen Jay</a> (1981). <a href="/wiki/The_Mismeasure_of_Man" title="The Mismeasure of Man"><i>The Mismeasure of Man</i></a> (first&#160;ed.). W. W. Norton. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-0-393-01489-1" title="Special:BookSources/978-0-393-01489-1"><bdi>978-0-393-01489-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=The+Mismeasure+of+Man&amp;rft.edition=first&amp;rft.pub=W.+W.+Norton&amp;rft.date=1981&amp;rft.isbn=978-0-393-01489-1&amp;rft.aulast=Gould&amp;rft.aufirst=Stephen+Jay&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFHalperinHartleyHoel1965" class="citation journal cs1">Halperin, Max; Hartley, Herman O.; Hoel, Paul G. (1965). "Recommended Standards for Statistical Symbols and Notation. COPSS Committee on Symbols and Notation". <i>The American Statistician</i>. <b>19</b> (3): 1214. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2681417">10.2307/2681417</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a>&#160;<a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2681417">2681417</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=The+American+Statistician&amp;rft.atitle=Recommended+Standards+for+Statistical+Symbols+and+Notation.+COPSS+Committee+on+Symbols+and+Notation&amp;rft.volume=19&amp;rft.issue=3&amp;rft.pages=12-14&amp;rft.date=1965&amp;rft_id=info%3Adoi%2F10.2307%2F2681417&amp;rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2681417%23id-name%3DJSTOR&amp;rft.aulast=Halperin&amp;rft.aufirst=Max&amp;rft.au=Hartley%2C+Herman+O.&amp;rft.au=Hoel%2C+Paul+G.&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFHart1968" class="citation book cs1">Hart, John F.; et&#160;al. (1968). <i>Computer Approximations</i>. New York, NY: John Wiley &amp; Sons, Inc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-0-88275-642-4" title="Special:BookSources/978-0-88275-642-4"><bdi>978-0-88275-642-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Computer+Approximations&amp;rft.place=New+York%2C+NY&amp;rft.pub=John+Wiley+%26+Sons%2C+Inc.&amp;rft.date=1968&amp;rft.isbn=978-0-88275-642-4&amp;rft.aulast=Hart&amp;rft.aufirst=John+F.&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Normal_Distribution">"Normal Distribution"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=bookitem&amp;rft.atitle=Normal+Distribution&amp;rft.btitle=Encyclopedia+of+Mathematics&amp;rft.pub=EMS+Press&amp;rft.date=2001&amp;rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DNormal_Distribution&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFHerrnsteinMurray1994" class="citation book cs1">Herrnstein, Richard J.; <a href="/wiki/Charles_Murray_(political_scientist)" title="Charles Murray (political scientist)">Murray, Charles</a> (1994). <a href="/wiki/The_Bell_Curve" title="The Bell Curve"><i>The Bell Curve: Intelligence and Class Structure in American Life</i></a>. <a href="/wiki/Free_Press_(publisher)" title="Free Press (publisher)">Free Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-0-02-914673-6" title="Special:BookSources/978-0-02-914673-6"><bdi>978-0-02-914673-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=The+Bell+Curve%3A+Intelligence+and+Class+Structure+in+American+Life&amp;rft.pub=Free+Press&amp;rft.date=1994&amp;rft.isbn=978-0-02-914673-6&amp;rft.aulast=Herrnstein&amp;rft.aufirst=Richard+J.&amp;rft.au=Murray%2C+Charles&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFHuxley1932" class="citation book cs1">Huxley, Julian S. (1932). <i>Problems of Relative Growth</i>. London. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-0-486-61114-3" title="Special:BookSources/978-0-486-61114-3"><bdi>978-0-486-61114-3</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a>&#160;<a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/476909537">476909537</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Problems+of+Relative+Growth&amp;rft.pub=London&amp;rft.date=1932&amp;rft_id=info%3Aoclcnum%2F476909537&amp;rft.isbn=978-0-486-61114-3&amp;rft.aulast=Huxley&amp;rft.aufirst=Julian+S.&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFJohnsonKotzBalakrishnan1994" class="citation book cs1">Johnson, Norman L.; Kotz, Samuel; Balakrishnan, Narayanaswamy (1994). <i>Continuous Univariate Distributions, Volume 1</i>. Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-0-471-58495-7" title="Special:BookSources/978-0-471-58495-7"><bdi>978-0-471-58495-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Continuous+Univariate+Distributions%2C+Volume+1&amp;rft.pub=Wiley&amp;rft.date=1994&amp;rft.isbn=978-0-471-58495-7&amp;rft.aulast=Johnson&amp;rft.aufirst=Norman+L.&amp;rft.au=Kotz%2C+Samuel&amp;rft.au=Balakrishnan%2C+Narayanaswamy&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFJohnsonKotzBalakrishnan1995" class="citation book cs1">Johnson, Norman L.; Kotz, Samuel; Balakrishnan, Narayanaswamy (1995). <i>Continuous Univariate Distributions, Volume 2</i>. Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-0-471-58494-0" title="Special:BookSources/978-0-471-58494-0"><bdi>978-0-471-58494-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Continuous+Univariate+Distributions%2C+Volume+2&amp;rft.pub=Wiley&amp;rft.date=1995&amp;rft.isbn=978-0-471-58494-0&amp;rft.aulast=Johnson&amp;rft.aufirst=Norman+L.&amp;rft.au=Kotz%2C+Samuel&amp;rft.au=Balakrishnan%2C+Narayanaswamy&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFKarney2016" class="citation journal cs1">Karney, C. F. F. (2016). "Sampling exactly from the normal distribution". <i>ACM Transactions on Mathematical Software</i>. <b>42</b> (1): 3:114. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="cs1-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1303.6257">1303.6257</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F2710016">10.1145/2710016</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a>&#160;<a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:14252035">14252035</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=ACM+Transactions+on+Mathematical+Software&amp;rft.atitle=Sampling+exactly+from+the+normal+distribution&amp;rft.volume=42&amp;rft.issue=1&amp;rft.pages=3%3A1-14&amp;rft.date=2016&amp;rft_id=info%3Aarxiv%2F1303.6257&amp;rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A14252035%23id-name%3DS2CID&amp;rft_id=info%3Adoi%2F10.1145%2F2710016&amp;rft.aulast=Karney&amp;rft.aufirst=C.+F.+F.&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFKindermanMonahan1977" class="citation journal cs1">Kinderman, Albert J.; Monahan, John F. (1977). <a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F355744.355750">"Computer Generation of Random Variables Using the Ratio of Uniform Deviates"</a>. <i>ACM Transactions on Mathematical Software</i>. <b>3</b> (3): 257260. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="cs1-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F355744.355750">10.1145/355744.355750</a></span>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a>&#160;<a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:12884505">12884505</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=ACM+Transactions+on+Mathematical+Software&amp;rft.atitle=Computer+Generation+of+Random+Variables+Using+the+Ratio+of+Uniform+Deviates&amp;rft.volume=3&amp;rft.issue=3&amp;rft.pages=257-260&amp;rft.date=1977&amp;rft_id=info%3Adoi%2F10.1145%2F355744.355750&amp;rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A12884505%23id-name%3DS2CID&amp;rft.aulast=Kinderman&amp;rft.aufirst=Albert+J.&amp;rft.au=Monahan%2C+John+F.&amp;rft_id=https%3A%2F%2Fdoi.org%2F10.1145%252F355744.355750&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFKrishnamoorthy2006" class="citation book cs1">Krishnamoorthy, Kalimuthu (2006). <i>Handbook of Statistical Distributions with Applications</i>. Chapman &amp; Hall/CRC. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-1-58488-635-8" title="Special:BookSources/978-1-58488-635-8"><bdi>978-1-58488-635-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Handbook+of+Statistical+Distributions+with+Applications&amp;rft.pub=Chapman+%26+Hall%2FCRC&amp;rft.date=2006&amp;rft.isbn=978-1-58488-635-8&amp;rft.aulast=Krishnamoorthy&amp;rft.aufirst=Kalimuthu&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFKruskalStigler1997" class="citation book cs1">Kruskal, William H.; Stigler, Stephen M. (1997). Spencer, Bruce D. (ed.). <i>Normative Terminology: 'Normal' in Statistics and Elsewhere</i>. Statistics and Public Policy. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-0-19-852341-3" title="Special:BookSources/978-0-19-852341-3"><bdi>978-0-19-852341-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Normative+Terminology%3A+%27Normal%27+in+Statistics+and+Elsewhere&amp;rft.series=Statistics+and+Public+Policy&amp;rft.pub=Oxford+University+Press&amp;rft.date=1997&amp;rft.isbn=978-0-19-852341-3&amp;rft.aulast=Kruskal&amp;rft.aufirst=William+H.&amp;rft.au=Stigler%2C+Stephen+M.&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFLaplace1774" class="citation journal cs1"><a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Laplace, Pierre-Simon de</a> (1774). <a rel="nofollow" class="external text" href="http://gallica.bnf.fr/ark:/12148/bpt6k77596b/f32">"Mémoire sur la probabilité des causes par les événements"</a>. <i>Mémoires de l'Académie Royale des Sciences de Paris (Savants étrangers), Tome 6</i>: 621656.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=M%C3%A9moires+de+l%27Acad%C3%A9mie+Royale+des+Sciences+de+Paris+%28Savants+%C3%A9trangers%29%2C+Tome+6&amp;rft.atitle=M%C3%A9moire+sur+la+probabilit%C3%A9+des+causes+par+les+%C3%A9v%C3%A9nements&amp;rft.pages=621-656&amp;rft.date=1774&amp;rft.aulast=Laplace&amp;rft.aufirst=Pierre-Simon+de&amp;rft_id=http%3A%2F%2Fgallica.bnf.fr%2Fark%3A%2F12148%2Fbpt6k77596b%2Ff32&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span> Translated by Stephen M. Stigler in <i>Statistical Science</i> <b>1</b> (3), 1986: <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a>&#160;<a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2245476">2245476</a>.</li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFLaplace1812" class="citation book cs1">Laplace, Pierre-Simon (1812). <a rel="nofollow" class="external text" href="https://archive.org/details/thorieanalytiqu00laplgoog"><i>Théorie analytique des probabilités</i></a> &#91;<i><a href="/wiki/Analytical_theory_of_probabilities" class="mw-redirect" title="Analytical theory of probabilities">Analytical theory of probabilities</a></i>&#93;. Paris, Ve. Courcier.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Th%C3%A9orie+analytique+des+probabilit%C3%A9s&amp;rft.pub=Paris%2C+Ve.+Courcier&amp;rft.date=1812&amp;rft.aulast=Laplace&amp;rft.aufirst=Pierre-Simon&amp;rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fthorieanalytiqu00laplgoog&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFLe_CamLo_Yang2000" class="citation book cs1">Le Cam, Lucien; Lo Yang, Grace (2000). <i>Asymptotics in Statistics: Some Basic Concepts</i> (second&#160;ed.). Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-0-387-95036-5" title="Special:BookSources/978-0-387-95036-5"><bdi>978-0-387-95036-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Asymptotics+in+Statistics%3A+Some+Basic+Concepts&amp;rft.edition=second&amp;rft.pub=Springer&amp;rft.date=2000&amp;rft.isbn=978-0-387-95036-5&amp;rft.aulast=Le+Cam&amp;rft.aufirst=Lucien&amp;rft.au=Lo+Yang%2C+Grace&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFLeva1992" class="citation journal cs1">Leva, Joseph L. (1992). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100716035328/http://saluc.engr.uconn.edu/refs/crypto/rng/leva92afast.pdf">"A fast normal random number generator"</a> <span class="cs1-format">(PDF)</span>. <i>ACM Transactions on Mathematical Software</i>. <b>18</b> (4): 449453. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a>&#160;<span class="cs1-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.544.5806">10.1.1.544.5806</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F138351.138364">10.1145/138351.138364</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a>&#160;<a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:15802663">15802663</a>. Archived from <a rel="nofollow" class="external text" href="http://saluc.engr.uconn.edu/refs/crypto/rng/leva92afast.pdf">the original</a> <span class="cs1-format">(PDF)</span> on July 16, 2010.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=ACM+Transactions+on+Mathematical+Software&amp;rft.atitle=A+fast+normal+random+number+generator&amp;rft.volume=18&amp;rft.issue=4&amp;rft.pages=449-453&amp;rft.date=1992&amp;rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.544.5806%23id-name%3DCiteSeerX&amp;rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A15802663%23id-name%3DS2CID&amp;rft_id=info%3Adoi%2F10.1145%2F138351.138364&amp;rft.aulast=Leva&amp;rft.aufirst=Joseph+L.&amp;rft_id=http%3A%2F%2Fsaluc.engr.uconn.edu%2Frefs%2Fcrypto%2Frng%2Fleva92afast.pdf&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFLexis1878" class="citation journal cs1">Lexis, Wilhelm (1878). "Sur la durée normale de la vie humaine et sur la théorie de la stabilité des rapports statistiques". <i>Annales de Démographie Internationale</i>. Paris. <b>II</b>: 447462.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Annales+de+D%C3%A9mographie+Internationale&amp;rft.atitle=Sur+la+dur%C3%A9e+normale+de+la+vie+humaine+et+sur+la+th%C3%A9orie+de+la+stabilit%C3%A9+des+rapports+statistiques&amp;rft.volume=II&amp;rft.pages=447-462&amp;rft.date=1878&amp;rft.aulast=Lexis&amp;rft.aufirst=Wilhelm&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFLukacsKing1954" class="citation journal cs1">Lukacs, Eugene; King, Edgar P. (1954). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faoms%2F1177728796">"A Property of Normal Distribution"</a>. <i>The Annals of Mathematical Statistics</i>. <b>25</b> (2): 389394. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="cs1-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faoms%2F1177728796">10.1214/aoms/1177728796</a></span>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a>&#160;<a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2236741">2236741</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=The+Annals+of+Mathematical+Statistics&amp;rft.atitle=A+Property+of+Normal+Distribution&amp;rft.volume=25&amp;rft.issue=2&amp;rft.pages=389-394&amp;rft.date=1954&amp;rft_id=info%3Adoi%2F10.1214%2Faoms%2F1177728796&amp;rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2236741%23id-name%3DJSTOR&amp;rft.aulast=Lukacs&amp;rft.aufirst=Eugene&amp;rft.au=King%2C+Edgar+P.&amp;rft_id=https%3A%2F%2Fdoi.org%2F10.1214%252Faoms%252F1177728796&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFMcPherson1990" class="citation book cs1">McPherson, Glen (1990). <span class="cs1-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/statisticsinscie0000mcph"><i>Statistics in Scientific Investigation: Its Basis, Application and Interpretation</i></a></span>. Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-0-387-97137-7" title="Special:BookSources/978-0-387-97137-7"><bdi>978-0-387-97137-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Statistics+in+Scientific+Investigation%3A+Its+Basis%2C+Application+and+Interpretation&amp;rft.pub=Springer-Verlag&amp;rft.date=1990&amp;rft.isbn=978-0-387-97137-7&amp;rft.aulast=McPherson&amp;rft.aufirst=Glen&amp;rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fstatisticsinscie0000mcph&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFMarsagliaTsang2000" class="citation journal cs1"><a href="/wiki/George_Marsaglia" title="George Marsaglia">Marsaglia, George</a>; Tsang, Wai Wan (2000). <a rel="nofollow" class="external text" href="https://doi.org/10.18637%2Fjss.v005.i08">"The Ziggurat Method for Generating Random Variables"</a>. <i>Journal of Statistical Software</i>. <b>5</b> (8). <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="cs1-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.18637%2Fjss.v005.i08">10.18637/jss.v005.i08</a></span>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Journal+of+Statistical+Software&amp;rft.atitle=The+Ziggurat+Method+for+Generating+Random+Variables&amp;rft.volume=5&amp;rft.issue=8&amp;rft.date=2000&amp;rft_id=info%3Adoi%2F10.18637%2Fjss.v005.i08&amp;rft.aulast=Marsaglia&amp;rft.aufirst=George&amp;rft.au=Tsang%2C+Wai+Wan&amp;rft_id=https%3A%2F%2Fdoi.org%2F10.18637%252Fjss.v005.i08&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFMarsaglia2004" class="citation journal cs1">Marsaglia, George (2004). <a rel="nofollow" class="external text" href="https://doi.org/10.18637%2Fjss.v011.i04">"Evaluating the Normal Distribution"</a>. <i>Journal of Statistical Software</i>. <b>11</b> (4). <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="cs1-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.18637%2Fjss.v011.i04">10.18637/jss.v011.i04</a></span>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Journal+of+Statistical+Software&amp;rft.atitle=Evaluating+the+Normal+Distribution&amp;rft.volume=11&amp;rft.issue=4&amp;rft.date=2004&amp;rft_id=info%3Adoi%2F10.18637%2Fjss.v011.i04&amp;rft.aulast=Marsaglia&amp;rft.aufirst=George&amp;rft_id=https%3A%2F%2Fdoi.org%2F10.18637%252Fjss.v011.i04&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFMaxwell1860" class="citation journal cs1"><a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell, James Clerk</a> (1860). "V. Illustrations of the dynamical theory of gases. — Part I: On the motions and collisions of perfectly elastic spheres". <i>Philosophical Magazine</i>. Series 4. <b>19</b> (124): 1932. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F14786446008642818">10.1080/14786446008642818</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Philosophical+Magazine&amp;rft.atitle=V.+Illustrations+of+the+dynamical+theory+of+gases.+%E2%80%94+Part+I%3A+On+the+motions+and+collisions+of+perfectly+elastic+spheres&amp;rft.volume=19&amp;rft.issue=124&amp;rft.pages=19-32&amp;rft.date=1860&amp;rft_id=info%3Adoi%2F10.1080%2F14786446008642818&amp;rft.aulast=Maxwell&amp;rft.aufirst=James+Clerk&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFMonahan1985" class="citation journal cs1">Monahan, J. F. (1985). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0025-5718-1985-0804945-X">"Accuracy in random number generation"</a>. <i>Mathematics of Computation</i>. <b>45</b> (172): 559568. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="cs1-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0025-5718-1985-0804945-X">10.1090/S0025-5718-1985-0804945-X</a></span>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Mathematics+of+Computation&amp;rft.atitle=Accuracy+in+random+number+generation&amp;rft.volume=45&amp;rft.issue=172&amp;rft.pages=559-568&amp;rft.date=1985&amp;rft_id=info%3Adoi%2F10.1090%2FS0025-5718-1985-0804945-X&amp;rft.aulast=Monahan&amp;rft.aufirst=J.+F.&amp;rft_id=https%3A%2F%2Fdoi.org%2F10.1090%252FS0025-5718-1985-0804945-X&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFPatelRead1996" class="citation book cs1">Patel, Jagdish K.; Read, Campbell B. (1996). <i>Handbook of the Normal Distribution</i> (2nd&#160;ed.). CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-0-8247-9342-5" title="Special:BookSources/978-0-8247-9342-5"><bdi>978-0-8247-9342-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Handbook+of+the+Normal+Distribution&amp;rft.edition=2nd&amp;rft.pub=CRC+Press&amp;rft.date=1996&amp;rft.isbn=978-0-8247-9342-5&amp;rft.aulast=Patel&amp;rft.aufirst=Jagdish+K.&amp;rft.au=Read%2C+Campbell+B.&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
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<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFPearson1905" class="citation journal cs1"><a href="/wiki/Karl_Pearson" title="Karl Pearson">Pearson, Karl</a> (1905). <a rel="nofollow" class="external text" href="https://zenodo.org/record/1449456">"<span class="cs1-kern-left"></span>'Das Fehlergesetz und seine Verallgemeinerungen durch Fechner und Pearson'. A rejoinder"</a>. <i>Biometrika</i>. <b>4</b> (1): 169212. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2331536">10.2307/2331536</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a>&#160;<a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2331536">2331536</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Biometrika&amp;rft.atitle=%27Das+Fehlergesetz+und+seine+Verallgemeinerungen+durch+Fechner+und+Pearson%27.+A+rejoinder&amp;rft.volume=4&amp;rft.issue=1&amp;rft.pages=169-212&amp;rft.date=1905&amp;rft_id=info%3Adoi%2F10.2307%2F2331536&amp;rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2331536%23id-name%3DJSTOR&amp;rft.aulast=Pearson&amp;rft.aufirst=Karl&amp;rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F1449456&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
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<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFRohrbasserVéron2003" class="citation journal cs1">Rohrbasser, Jean-Marc; Véron, Jacques (2003). <a rel="nofollow" class="external text" href="http://www.persee.fr/web/revues/home/prescript/article/pop_1634-2941_2003_num_58_3_18444">"Wilhelm Lexis: The Normal Length of Life as an Expression of the "Nature of Things"<span class="cs1-kern-right"></span>"</a>. <i>Population</i>. <b>58</b> (3): 303322. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="cs1-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.3917%2Fpope.303.0303">10.3917/pope.303.0303</a></span>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Population&amp;rft.atitle=Wilhelm+Lexis%3A+The+Normal+Length+of+Life+as+an+Expression+of+the+%22Nature+of+Things%22&amp;rft.volume=58&amp;rft.issue=3&amp;rft.pages=303-322&amp;rft.date=2003&amp;rft_id=info%3Adoi%2F10.3917%2Fpope.303.0303&amp;rft.aulast=Rohrbasser&amp;rft.aufirst=Jean-Marc&amp;rft.au=V%C3%A9ron%2C+Jacques&amp;rft_id=http%3A%2F%2Fwww.persee.fr%2Fweb%2Frevues%2Fhome%2Fprescript%2Farticle%2Fpop_1634-2941_2003_num_58_3_18444&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
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<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFShore2005" class="citation journal cs1">Shore, H (2005). "Accurate RMM-Based Approximations for the CDF of the Normal Distribution". <i>Communications in Statistics Theory and Methods</i>. <b>34</b> (3): 507513. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1081%2Fsta-200052102">10.1081/sta-200052102</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a>&#160;<a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122148043">122148043</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Communications+in+Statistics+%E2%80%93+Theory+and+Methods&amp;rft.atitle=Accurate+RMM-Based+Approximations+for+the+CDF+of+the+Normal+Distribution&amp;rft.volume=34&amp;rft.issue=3&amp;rft.pages=507-513&amp;rft.date=2005&amp;rft_id=info%3Adoi%2F10.1081%2Fsta-200052102&amp;rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122148043%23id-name%3DS2CID&amp;rft.aulast=Shore&amp;rft.aufirst=H&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFShore2011" class="citation journal cs1">Shore, H (2011). "Response Modeling Methodology". <i>WIREs Comput Stat</i>. <b>3</b> (4): 357372. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fwics.151">10.1002/wics.151</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a>&#160;<a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:62021374">62021374</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=WIREs+Comput+Stat&amp;rft.atitle=Response+Modeling+Methodology&amp;rft.volume=3&amp;rft.issue=4&amp;rft.pages=357-372&amp;rft.date=2011&amp;rft_id=info%3Adoi%2F10.1002%2Fwics.151&amp;rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A62021374%23id-name%3DS2CID&amp;rft.aulast=Shore&amp;rft.aufirst=H&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFShore2012" class="citation journal cs1">Shore, H (2012). "Estimating Response Modeling Methodology Models". <i>WIREs Comput Stat</i>. <b>4</b> (3): 323333. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fwics.1199">10.1002/wics.1199</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a>&#160;<a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122366147">122366147</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=WIREs+Comput+Stat&amp;rft.atitle=Estimating+Response+Modeling+Methodology+Models&amp;rft.volume=4&amp;rft.issue=3&amp;rft.pages=323-333&amp;rft.date=2012&amp;rft_id=info%3Adoi%2F10.1002%2Fwics.1199&amp;rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122366147%23id-name%3DS2CID&amp;rft.aulast=Shore&amp;rft.aufirst=H&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
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<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFStigler1982" class="citation journal cs1">Stigler, Stephen M. (1982). "A Modest Proposal: A New Standard for the Normal". <i>The American Statistician</i>. <b>36</b> (2): 137138. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2684031">10.2307/2684031</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a>&#160;<a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2684031">2684031</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=The+American+Statistician&amp;rft.atitle=A+Modest+Proposal%3A+A+New+Standard+for+the+Normal&amp;rft.volume=36&amp;rft.issue=2&amp;rft.pages=137-138&amp;rft.date=1982&amp;rft_id=info%3Adoi%2F10.2307%2F2684031&amp;rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2684031%23id-name%3DJSTOR&amp;rft.aulast=Stigler&amp;rft.aufirst=Stephen+M.&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
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<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFWallace1996" class="citation journal cs1"><a href="/wiki/Chris_Wallace_(computer_scientist)" title="Chris Wallace (computer scientist)">Wallace, C. S.</a> (1996). <a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F225545.225554">"Fast pseudo-random generators for normal and exponential variates"</a>. <i>ACM Transactions on Mathematical Software</i>. <b>22</b> (1): 119127. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="cs1-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F225545.225554">10.1145/225545.225554</a></span>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a>&#160;<a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:18514848">18514848</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=ACM+Transactions+on+Mathematical+Software&amp;rft.atitle=Fast+pseudo-random+generators+for+normal+and+exponential+variates&amp;rft.volume=22&amp;rft.issue=1&amp;rft.pages=119-127&amp;rft.date=1996&amp;rft_id=info%3Adoi%2F10.1145%2F225545.225554&amp;rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A18514848%23id-name%3DS2CID&amp;rft.aulast=Wallace&amp;rft.aufirst=C.+S.&amp;rft_id=https%3A%2F%2Fdoi.org%2F10.1145%252F225545.225554&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/NormalDistribution.html">"Normal Distribution"</a>. <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=unknown&amp;rft.btitle=Normal+Distribution&amp;rft.pub=MathWorld&amp;rft.aulast=Weisstein&amp;rft.aufirst=Eric+W.&amp;rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FNormalDistribution.html&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFWest2009" class="citation journal cs1">West, Graeme (2009). <a rel="nofollow" class="external text" href="http://www.wilmott.com/pdfs/090721_west.pdf">"Better Approximations to Cumulative Normal Functions"</a> <span class="cs1-format">(PDF)</span>. <i>Wilmott Magazine</i>: 7076.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Wilmott+Magazine&amp;rft.atitle=Better+Approximations+to+Cumulative+Normal+Functions&amp;rft.pages=70-76&amp;rft.date=2009&amp;rft.aulast=West&amp;rft.aufirst=Graeme&amp;rft_id=http%3A%2F%2Fwww.wilmott.com%2Fpdfs%2F090721_west.pdf&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite id="CITEREFZelenSevero1964" class="citation book cs1">Zelen, Marvin; Severo, Norman C. (1964). <a rel="nofollow" class="external text" href="http://www.math.sfu.ca/~cbm/aands/page_931.htm"><i>Probability Functions (chapter 26)</i></a>. <i><a href="/wiki/Abramowitz_and_Stegun" title="Abramowitz and Stegun">Handbook of mathematical functions with formulas, graphs, and mathematical tables</a></i>, by <a href="/wiki/Milton_Abramowitz" title="Milton Abramowitz">Abramowitz, M.</a>; and <a href="/wiki/Irene_A._Stegun" class="mw-redirect" title="Irene A. Stegun">Stegun, I. A.</a>: National Bureau of Standards. New York, NY: Dover. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-0-486-61272-0" title="Special:BookSources/978-0-486-61272-0"><bdi>978-0-486-61272-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Probability+Functions+%28chapter+26%29&amp;rft.place=New+York%2C+NY&amp;rft.series=%27%27Handbook+of+mathematical+functions+with+formulas%2C+graphs%2C+and+mathematical+tables%27%27%2C+by+Abramowitz%2C+M.%3B+and+Stegun%2C+I.+A.%3A+National+Bureau+of+Standards&amp;rft.pub=Dover&amp;rft.date=1964&amp;rft.isbn=978-0-486-61272-0&amp;rft.aulast=Zelen&amp;rft.aufirst=Marvin&amp;rft.au=Severo%2C+Norman+C.&amp;rft_id=http%3A%2F%2Fwww.math.sfu.ca%2F~cbm%2Faands%2Fpage_931.htm&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li></ul>
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<h2><span class="mw-headline" id="External_links">External links</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_distribution&amp;action=edit&amp;section=63" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></h2>
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<ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1133582631"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Normal_distribution">"Normal distribution"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=bookitem&amp;rft.atitle=Normal+distribution&amp;rft.btitle=Encyclopedia+of+Mathematics&amp;rft.pub=EMS+Press&amp;rft.date=2001&amp;rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DNormal_distribution&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+distribution" class="Z3988"></span></li>
<li><a rel="nofollow" class="external text" href="https://www.hackmath.net/en/calculator/normal-distribution">Normal distribution calculator</a></li></ul>
<div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1061467846">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}</style><style data-mw-deduplicate="TemplateStyles:r886047488">.mw-parser-output .nobold{font-weight:normal}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886047488"></div><div role="navigation" class="navbox" aria-labelledby="Probability_distributions_(list)" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1063604349"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Probability_distributions" title="Template:Probability distributions"><abbr title="View this template" style=";;background:none transparent;border:none;box-shadow:none;padding:0;">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Probability_distributions" title="Template talk:Probability distributions"><abbr title="Discuss this template" style=";;background:none transparent;border:none;box-shadow:none;padding:0;">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Probability_distributions" title="Special:EditPage/Template:Probability distributions"><abbr title="Edit this template" style=";;background:none transparent;border:none;box-shadow:none;p
<ul><li><a href="/wiki/Benford%27s_law" title="Benford&#39;s law">Benford</a></li>
<li><a href="/wiki/Bernoulli_distribution" title="Bernoulli distribution">Bernoulli</a></li>
<li><a href="/wiki/Beta-binomial_distribution" title="Beta-binomial distribution">beta-binomial</a></li>
<li><a href="/wiki/Binomial_distribution" title="Binomial distribution">binomial</a></li>
<li><a href="/wiki/Categorical_distribution" title="Categorical distribution">categorical</a></li>
<li><a href="/wiki/Hypergeometric_distribution" title="Hypergeometric distribution">hypergeometric</a>
<ul><li><a href="/wiki/Negative_hypergeometric_distribution" title="Negative hypergeometric distribution">negative</a></li></ul></li>
<li><a href="/wiki/Poisson_binomial_distribution" title="Poisson binomial distribution">Poisson binomial</a></li>
<li><a href="/wiki/Rademacher_distribution" title="Rademacher distribution">Rademacher</a></li>
<li><a href="/wiki/Soliton_distribution" title="Soliton distribution">soliton</a></li>
<li><a href="/wiki/Discrete_uniform_distribution" title="Discrete uniform distribution">discrete uniform</a></li>
<li><a href="/wiki/Zipf%27s_law" title="Zipf&#39;s law">Zipf</a></li>
<li><a href="/wiki/Zipf%E2%80%93Mandelbrot_law" title="ZipfMandelbrot law">ZipfMandelbrot</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">with infinite <br />support</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="/wiki/Beta_negative_binomial_distribution" title="Beta negative binomial distribution">beta negative binomial</a></li>
<li><a href="/wiki/Borel_distribution" title="Borel distribution">Borel</a></li>
<li><a href="/wiki/Conway%E2%80%93Maxwell%E2%80%93Poisson_distribution" title="ConwayMaxwellPoisson distribution">ConwayMaxwellPoisson</a></li>
<li><a href="/wiki/Discrete_phase-type_distribution" title="Discrete phase-type distribution">discrete phase-type</a></li>
<li><a href="/wiki/Delaporte_distribution" title="Delaporte distribution">Delaporte</a></li>
<li><a href="/wiki/Extended_negative_binomial_distribution" title="Extended negative binomial distribution">extended negative binomial</a></li>
<li><a href="/wiki/Flory%E2%80%93Schulz_distribution" title="FlorySchulz distribution">FlorySchulz</a></li>
<li><a href="/wiki/Gauss%E2%80%93Kuzmin_distribution" title="GaussKuzmin distribution">GaussKuzmin</a></li>
<li><a href="/wiki/Geometric_distribution" title="Geometric distribution">geometric</a></li>
<li><a href="/wiki/Logarithmic_distribution" title="Logarithmic distribution">logarithmic</a></li>
<li><a href="/wiki/Mixed_Poisson_distribution" title="Mixed Poisson distribution">mixed Poisson</a></li>
<li><a href="/wiki/Negative_binomial_distribution" title="Negative binomial distribution">negative binomial</a></li>
<li><a href="/wiki/(a,b,0)_class_of_distributions" title="(a,b,0) class of distributions">Panjer</a></li>
<li><a href="/wiki/Parabolic_fractal_distribution" title="Parabolic fractal distribution">parabolic fractal</a></li>
<li><a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson</a></li>
<li><a href="/wiki/Skellam_distribution" title="Skellam distribution">Skellam</a></li>
<li><a href="/wiki/Yule%E2%80%93Simon_distribution" title="YuleSimon distribution">YuleSimon</a></li>
<li><a href="/wiki/Zeta_distribution" title="Zeta distribution">zeta</a></li></ul>
</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Continuous <br />univariate</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">supported on a <br />bounded interval</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="/wiki/Arcsine_distribution" title="Arcsine distribution">arcsine</a></li>
<li><a href="/wiki/ARGUS_distribution" title="ARGUS distribution">ARGUS</a></li>
<li><a href="/wiki/Balding%E2%80%93Nichols_model" title="BaldingNichols model">BaldingNichols</a></li>
<li><a href="/wiki/Bates_distribution" title="Bates distribution">Bates</a></li>
<li><a href="/wiki/Beta_distribution" title="Beta distribution">beta</a></li>
<li><a href="/wiki/Beta_rectangular_distribution" title="Beta rectangular distribution">beta rectangular</a></li>
<li><a href="/wiki/Continuous_Bernoulli_distribution" title="Continuous Bernoulli distribution">continuous Bernoulli</a></li>
<li><a href="/wiki/Irwin%E2%80%93Hall_distribution" title="IrwinHall distribution">IrwinHall</a></li>
<li><a href="/wiki/Kumaraswamy_distribution" title="Kumaraswamy distribution">Kumaraswamy</a></li>
<li><a href="/wiki/Logit-normal_distribution" title="Logit-normal distribution">logit-normal</a></li>
<li><a href="/wiki/Noncentral_beta_distribution" title="Noncentral beta distribution">noncentral beta</a></li>
<li><a href="/wiki/PERT_distribution" title="PERT distribution">PERT</a></li>
<li><a href="/wiki/Raised_cosine_distribution" title="Raised cosine distribution">raised cosine</a></li>
<li><a href="/wiki/Reciprocal_distribution" title="Reciprocal distribution">reciprocal</a></li>
<li><a href="/wiki/Triangular_distribution" title="Triangular distribution">triangular</a></li>
<li><a href="/wiki/U-quadratic_distribution" title="U-quadratic distribution">U-quadratic</a></li>
<li><a href="/wiki/Continuous_uniform_distribution" title="Continuous uniform distribution">uniform</a></li>
<li><a href="/wiki/Wigner_semicircle_distribution" title="Wigner semicircle distribution">Wigner semicircle</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">supported on a <br />semi-infinite <br />interval</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="/wiki/Benini_distribution" title="Benini distribution">Benini</a></li>
<li><a href="/wiki/Benktander_type_I_distribution" title="Benktander type I distribution">Benktander 1st kind</a></li>
<li><a href="/wiki/Benktander_type_II_distribution" title="Benktander type II distribution">Benktander 2nd kind</a></li>
<li><a href="/wiki/Beta_prime_distribution" title="Beta prime distribution">beta prime</a></li>
<li><a href="/wiki/Burr_distribution" title="Burr distribution">Burr</a></li>
<li><a href="/wiki/Chi_distribution" title="Chi distribution">chi</a></li>
<li><a href="/wiki/Chi-squared_distribution" title="Chi-squared distribution">chi-squared</a>
<ul><li><a href="/wiki/Noncentral_chi-squared_distribution" title="Noncentral chi-squared distribution">noncentral</a></li>
<li><a href="/wiki/Inverse-chi-squared_distribution" title="Inverse-chi-squared distribution">inverse</a>
<ul><li><a href="/wiki/Scaled_inverse_chi-squared_distribution" title="Scaled inverse chi-squared distribution">scaled</a></li></ul></li></ul></li>
<li><a href="/wiki/Dagum_distribution" title="Dagum distribution">Dagum</a></li>
<li><a href="/wiki/Davis_distribution" title="Davis distribution">Davis</a></li>
<li><a href="/wiki/Erlang_distribution" title="Erlang distribution">Erlang</a>
<ul><li><a href="/wiki/Hyper-Erlang_distribution" title="Hyper-Erlang distribution">hyper</a></li></ul></li>
<li><a href="/wiki/Exponential_distribution" title="Exponential distribution">exponential</a>
<ul><li><a href="/wiki/Hyperexponential_distribution" title="Hyperexponential distribution">hyperexponential</a></li>
<li><a href="/wiki/Hypoexponential_distribution" title="Hypoexponential distribution">hypoexponential</a></li>
<li><a href="/wiki/Exponential-logarithmic_distribution" title="Exponential-logarithmic distribution">logarithmic</a></li></ul></li>
<li><a href="/wiki/F-distribution" title="F-distribution"><i>F</i></a>
<ul><li><a href="/wiki/Noncentral_F-distribution" title="Noncentral F-distribution">noncentral</a></li></ul></li>
<li><a href="/wiki/Folded_normal_distribution" title="Folded normal distribution">folded normal</a></li>
<li><a href="/wiki/Fr%C3%A9chet_distribution" title="Fréchet distribution">Fréchet</a></li>
<li><a href="/wiki/Gamma_distribution" title="Gamma distribution">gamma</a>
<ul><li><a href="/wiki/Generalized_gamma_distribution" title="Generalized gamma distribution">generalized</a></li>
<li><a href="/wiki/Inverse-gamma_distribution" title="Inverse-gamma distribution">inverse</a></li></ul></li>
<li><a href="/wiki/Gamma/Gompertz_distribution" title="Gamma/Gompertz distribution">gamma/Gompertz</a></li>
<li><a href="/wiki/Gompertz_distribution" title="Gompertz distribution">Gompertz</a>
<ul><li><a href="/wiki/Shifted_Gompertz_distribution" title="Shifted Gompertz distribution">shifted</a></li></ul></li>
<li><a href="/wiki/Half-logistic_distribution" title="Half-logistic distribution">half-logistic</a></li>
<li><a href="/wiki/Half-normal_distribution" title="Half-normal distribution">half-normal</a></li>
<li><a href="/wiki/Hotelling%27s_T-squared_distribution" title="Hotelling&#39;s T-squared distribution">Hotelling's <i>T</i>-squared</a></li>
<li><a href="/wiki/Inverse_Gaussian_distribution" title="Inverse Gaussian distribution">inverse Gaussian</a>
<ul><li><a href="/wiki/Generalized_inverse_Gaussian_distribution" title="Generalized inverse Gaussian distribution">generalized</a></li></ul></li>
<li><a href="/wiki/Kolmogorov%E2%80%93Smirnov_test" title="KolmogorovSmirnov test">Kolmogorov</a></li>
<li><a href="/wiki/L%C3%A9vy_distribution" title="Lévy distribution">Lévy</a></li>
<li><a href="/wiki/Log-Cauchy_distribution" title="Log-Cauchy distribution">log-Cauchy</a></li>
<li><a href="/wiki/Log-Laplace_distribution" title="Log-Laplace distribution">log-Laplace</a></li>
<li><a href="/wiki/Log-logistic_distribution" title="Log-logistic distribution">log-logistic</a></li>
<li><a href="/wiki/Log-normal_distribution" title="Log-normal distribution">log-normal</a></li>
<li><a href="/wiki/Log-t_distribution" title="Log-t distribution">log-t</a></li>
<li><a href="/wiki/Lomax_distribution" title="Lomax distribution">Lomax</a></li>
<li><a href="/wiki/Matrix-exponential_distribution" title="Matrix-exponential distribution">matrix-exponential</a></li>
<li><a href="/wiki/Maxwell%E2%80%93Boltzmann_distribution" title="MaxwellBoltzmann distribution">MaxwellBoltzmann</a></li>
<li><a href="/wiki/Maxwell%E2%80%93J%C3%BCttner_distribution" title="MaxwellJüttner distribution">MaxwellJüttner</a></li>
<li><a href="/wiki/Mittag-Leffler_distribution" title="Mittag-Leffler distribution">Mittag-Leffler</a></li>
<li><a href="/wiki/Nakagami_distribution" title="Nakagami distribution">Nakagami</a></li>
<li><a href="/wiki/Pareto_distribution" title="Pareto distribution">Pareto</a></li>
<li><a href="/wiki/Phase-type_distribution" title="Phase-type distribution">phase-type</a></li>
<li><a href="/wiki/Poly-Weibull_distribution" title="Poly-Weibull distribution">Poly-Weibull</a></li>
<li><a href="/wiki/Rayleigh_distribution" title="Rayleigh distribution">Rayleigh</a></li>
<li><a href="/wiki/Relativistic_Breit%E2%80%93Wigner_distribution" title="Relativistic BreitWigner distribution">relativistic BreitWigner</a></li>
<li><a href="/wiki/Rice_distribution" title="Rice distribution">Rice</a></li>
<li><a href="/wiki/Truncated_normal_distribution" title="Truncated normal distribution">truncated normal</a></li>
<li><a href="/wiki/Type-2_Gumbel_distribution" title="Type-2 Gumbel distribution">type-2 Gumbel</a></li>
<li><a href="/wiki/Weibull_distribution" title="Weibull distribution">Weibull</a>
<ul><li><a href="/wiki/Discrete_Weibull_distribution" title="Discrete Weibull distribution">discrete</a></li></ul></li>
<li><a href="/wiki/Wilks%27s_lambda_distribution" title="Wilks&#39;s lambda distribution">Wilks's lambda</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">supported <br />on the whole <br />real line</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="/wiki/Cauchy_distribution" title="Cauchy distribution">Cauchy</a></li>
<li><a href="/wiki/Generalized_normal_distribution#Version_1" title="Generalized normal distribution">exponential power</a></li>
<li><a href="/wiki/Fisher%27s_z-distribution" title="Fisher&#39;s z-distribution">Fisher's <i>z</i></a></li>
<li><a href="/wiki/Kaniadakis_Gaussian_distribution" title="Kaniadakis Gaussian distribution">Kaniadakis κ-Gaussian</a></li>
<li><a href="/wiki/Gaussian_q-distribution" title="Gaussian q-distribution">Gaussian <i>q</i></a></li>
<li><a href="/wiki/Generalized_normal_distribution" title="Generalized normal distribution">generalized normal</a></li>
<li><a href="/wiki/Generalised_hyperbolic_distribution" title="Generalised hyperbolic distribution">generalized hyperbolic</a></li>
<li><a href="/wiki/Geometric_stable_distribution" title="Geometric stable distribution">geometric stable</a></li>
<li><a href="/wiki/Gumbel_distribution" title="Gumbel distribution">Gumbel</a></li>
<li><a href="/wiki/Holtsmark_distribution" title="Holtsmark distribution">Holtsmark</a></li>
<li><a href="/wiki/Hyperbolic_secant_distribution" title="Hyperbolic secant distribution">hyperbolic secant</a></li>
<li><a href="/wiki/Johnson%27s_SU-distribution" title="Johnson&#39;s SU-distribution">Johnson's <i>S<sub>U</sub></i></a></li>
<li><a href="/wiki/Landau_distribution" title="Landau distribution">Landau</a></li>
<li><a href="/wiki/Laplace_distribution" title="Laplace distribution">Laplace</a>
<ul><li><a href="/wiki/Asymmetric_Laplace_distribution" title="Asymmetric Laplace distribution">asymmetric</a></li></ul></li>
<li><a href="/wiki/Logistic_distribution" title="Logistic distribution">logistic</a></li>
<li><a href="/wiki/Noncentral_t-distribution" title="Noncentral t-distribution">noncentral <i>t</i></a></li>
<li><a class="mw-selflink selflink">normal (Gaussian)</a></li>
<li><a href="/wiki/Normal-inverse_Gaussian_distribution" title="Normal-inverse Gaussian distribution">normal-inverse Gaussian</a></li>
<li><a href="/wiki/Skew_normal_distribution" title="Skew normal distribution">skew normal</a></li>
<li><a href="/wiki/Slash_distribution" title="Slash distribution">slash</a></li>
<li><a href="/wiki/Stable_distribution" title="Stable distribution">stable</a></li>
<li><a href="/wiki/Student%27s_t-distribution" title="Student&#39;s t-distribution">Student's <i>t</i></a></li>
<li><a href="/wiki/Tracy%E2%80%93Widom_distribution" title="TracyWidom distribution">TracyWidom</a></li>
<li><a href="/wiki/Variance-gamma_distribution" title="Variance-gamma distribution">variance-gamma</a></li>
<li><a href="/wiki/Voigt_profile" title="Voigt profile">Voigt</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">with support <br />whose type varies</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="/wiki/Generalized_chi-squared_distribution" title="Generalized chi-squared distribution">generalized chi-squared</a></li>
<li><a href="/wiki/Generalized_extreme_value_distribution" title="Generalized extreme value distribution">generalized extreme value</a></li>
<li><a href="/wiki/Generalized_Pareto_distribution" title="Generalized Pareto distribution">generalized Pareto</a></li>
<li><a href="/wiki/Marchenko%E2%80%93Pastur_distribution" title="MarchenkoPastur distribution">MarchenkoPastur</a></li>
<li><a href="/wiki/Kaniadakis_Exponential_distribution" class="mw-redirect" title="Kaniadakis Exponential distribution">Kaniadakis <i>κ</i>-exponential</a></li>
<li><a href="/wiki/Kaniadakis_Gamma_distribution" title="Kaniadakis Gamma distribution">Kaniadakis <i>κ</i>-Gamma</a></li>
<li><a href="/wiki/Kaniadakis_Weibull_distribution" title="Kaniadakis Weibull distribution">Kaniadakis <i>κ</i>-Weibull</a></li>
<li><a href="/wiki/Kaniadakis_Logistic_distribution" class="mw-redirect" title="Kaniadakis Logistic distribution">Kaniadakis <i>κ</i>-Logistic</a></li>
<li><a href="/wiki/Kaniadakis_Erlang_distribution" title="Kaniadakis Erlang distribution">Kaniadakis <i>κ</i>-Erlang</a></li>
<li><a href="/wiki/Q-exponential_distribution" title="Q-exponential distribution"><i>q</i>-exponential</a></li>
<li><a href="/wiki/Q-Gaussian_distribution" title="Q-Gaussian distribution"><i>q</i>-Gaussian</a></li>
<li><a href="/wiki/Q-Weibull_distribution" title="Q-Weibull distribution"><i>q</i>-Weibull</a></li>
<li><a href="/wiki/Shifted_log-logistic_distribution" title="Shifted log-logistic distribution">shifted log-logistic</a></li>
<li><a href="/wiki/Tukey_lambda_distribution" title="Tukey lambda distribution">Tukey lambda</a></li></ul>
</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Mixed <br />univariate</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">continuous-<br />discrete</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="/wiki/Rectified_Gaussian_distribution" title="Rectified Gaussian distribution">Rectified Gaussian</a></li></ul>
</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Joint_probability_distribution" title="Joint probability distribution">Multivariate <br />(joint)</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><span class="nobold"><i>Discrete: </i></span></li>
<li><a href="/wiki/Ewens%27s_sampling_formula" title="Ewens&#39;s sampling formula">Ewens</a></li>
<li><a href="/wiki/Multinomial_distribution" title="Multinomial distribution">multinomial</a>
<ul><li><a href="/wiki/Dirichlet-multinomial_distribution" title="Dirichlet-multinomial distribution">Dirichlet</a></li>
<li><a href="/wiki/Negative_multinomial_distribution" title="Negative multinomial distribution">negative</a></li></ul></li>
<li><span class="nobold"><i>Continuous: </i></span></li>
<li><a href="/wiki/Dirichlet_distribution" title="Dirichlet distribution">Dirichlet</a>
<ul><li><a href="/wiki/Generalized_Dirichlet_distribution" title="Generalized Dirichlet distribution">generalized</a></li></ul></li>
<li><a href="/wiki/Multivariate_Laplace_distribution" title="Multivariate Laplace distribution">multivariate Laplace</a></li>
<li><a href="/wiki/Multivariate_normal_distribution" title="Multivariate normal distribution">multivariate normal</a></li>
<li><a href="/wiki/Multivariate_stable_distribution" title="Multivariate stable distribution">multivariate stable</a></li>
<li><a href="/wiki/Multivariate_t-distribution" title="Multivariate t-distribution">multivariate <i>t</i></a></li>
<li><a href="/wiki/Normal-gamma_distribution" title="Normal-gamma distribution">normal-gamma</a>
<ul><li><a href="/wiki/Normal-inverse-gamma_distribution" title="Normal-inverse-gamma distribution">inverse</a></li></ul></li>
<li><span class="nobold"><i><a href="/wiki/Random_matrix" title="Random matrix">Matrix-valued: </a></i></span></li>
<li><a href="/wiki/Lewandowski-Kurowicka-Joe_distribution" title="Lewandowski-Kurowicka-Joe distribution">LKJ</a></li>
<li><a href="/wiki/Matrix_normal_distribution" title="Matrix normal distribution">matrix normal</a></li>
<li><a href="/wiki/Matrix_t-distribution" title="Matrix t-distribution">matrix <i>t</i></a></li>
<li><a href="/wiki/Matrix_gamma_distribution" title="Matrix gamma distribution">matrix gamma</a>
<ul><li><a href="/wiki/Inverse_matrix_gamma_distribution" title="Inverse matrix gamma distribution">inverse</a></li></ul></li>
<li><a href="/wiki/Wishart_distribution" title="Wishart distribution">Wishart</a>
<ul><li><a href="/wiki/Normal-Wishart_distribution" title="Normal-Wishart distribution">normal</a></li>
<li><a href="/wiki/Inverse-Wishart_distribution" title="Inverse-Wishart distribution">inverse</a></li>
<li><a href="/wiki/Normal-inverse-Wishart_distribution" title="Normal-inverse-Wishart distribution">normal-inverse</a></li>
<li><a href="/wiki/Complex_Wishart_distribution" title="Complex Wishart distribution">complex</a></li></ul></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Directional_statistics" title="Directional statistics">Directional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<dl><dt><span class="nobold"><i>Univariate (circular) <a href="/wiki/Directional_statistics" title="Directional statistics">directional</a></i></span></dt>
<dd><a href="/wiki/Circular_uniform_distribution" title="Circular uniform distribution">Circular uniform</a></dd>
<dd><a href="/wiki/Von_Mises_distribution" title="Von Mises distribution">univariate von Mises</a></dd>
<dd><a href="/wiki/Wrapped_normal_distribution" title="Wrapped normal distribution">wrapped normal</a></dd>
<dd><a href="/wiki/Wrapped_Cauchy_distribution" title="Wrapped Cauchy distribution">wrapped Cauchy</a></dd>
<dd><a href="/wiki/Wrapped_exponential_distribution" title="Wrapped exponential distribution">wrapped exponential</a></dd>
<dd><a href="/wiki/Wrapped_asymmetric_Laplace_distribution" title="Wrapped asymmetric Laplace distribution">wrapped asymmetric Laplace</a></dd>
<dd><a href="/wiki/Wrapped_L%C3%A9vy_distribution" title="Wrapped Lévy distribution">wrapped Lévy</a></dd>
<dt><span class="nobold"><i>Bivariate (spherical)</i></span></dt>
<dd><a href="/wiki/Kent_distribution" title="Kent distribution">Kent</a></dd>
<dt><span class="nobold"><i>Bivariate (toroidal)</i></span></dt>
<dd><a href="/wiki/Bivariate_von_Mises_distribution" title="Bivariate von Mises distribution">bivariate von Mises</a></dd>
<dt><span class="nobold"><i>Multivariate</i></span></dt>
<dd><a href="/wiki/Von_Mises%E2%80%93Fisher_distribution" title="Von MisesFisher distribution">von MisesFisher</a></dd>
<dd><a href="/wiki/Bingham_distribution" title="Bingham distribution">Bingham</a></dd></dl>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Degenerate_distribution" title="Degenerate distribution">Degenerate</a> <br />and <a href="/wiki/Singular_distribution" title="Singular distribution">singular</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<dl><dt><span class="nobold"><i>Degenerate</i></span></dt>
<dd><a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a></dd>
<dt><span class="nobold"><i>Singular</i></span></dt>
<dd><a href="/wiki/Cantor_distribution" title="Cantor distribution">Cantor</a></dd></dl>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Families</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="/wiki/Circular_distribution" title="Circular distribution">Circular</a></li>
<li><a href="/wiki/Compound_Poisson_distribution" title="Compound Poisson distribution">compound Poisson</a></li>
<li><a href="/wiki/Elliptical_distribution" title="Elliptical distribution">elliptical</a></li>
<li><a href="/wiki/Exponential_family" title="Exponential family">exponential</a></li>
<li><a href="/wiki/Natural_exponential_family" title="Natural exponential family">natural exponential</a></li>
<li><a href="/wiki/Location%E2%80%93scale_family" title="Locationscale family">locationscale</a></li>
<li><a href="/wiki/Maximum_entropy_probability_distribution" title="Maximum entropy probability distribution">maximum entropy</a></li>
<li><a href="/wiki/Mixture_distribution" title="Mixture distribution">mixture</a></li>
<li><a href="/wiki/Pearson_distribution" title="Pearson distribution">Pearson</a></li>
<li><a href="/wiki/Tweedie_distribution" title="Tweedie distribution">Tweedie</a></li>
<li><a href="/wiki/Wrapped_distribution" title="Wrapped distribution">wrapped</a></li></ul>
</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>
<ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Probability_distributions" title="Category:Probability distributions">Category</a></li>
<li><span class="noviewer" typeof="mw:File"><span title="Commons page"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span> <a href="https://commons.wikimedia.org/wiki/Category:Probability_distributions" class="extiw" title="commons:Category:Probability distributions">Commons</a></li></ul>
</div></td></tr></tbody></table></div>
<div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1061467846"><style data-mw-deduplicate="TemplateStyles:r1038841319">.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319"></div><div role="navigation" class="navbox authority-control" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a>: National <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q133871#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" src="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Gauss, Loi de (statistique)"><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb119421818">France</a></span></span></li>
<li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Gauss, Loi de (statistique)"><a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb119421818">BnF data</a></span></span></li>
<li><span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4075494-7">Germany</a></span></li>
<li><span class="uid"><a rel="nofollow" class="external text" href="http://uli.nli.org.il/F/?func=find-b&amp;local_base=NLX10&amp;find_code=UID&amp;request=987007560462505171">Israel</a></span></li>
<li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Gaussian distribution"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85053556">United States</a></span></span></li>
<li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="normální rozložení pravděpodobností"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&amp;local_base=aut&amp;ccl_term=ica=ph123321&amp;CON_LNG=ENG">Czech Republic</a></span></span></li></ul>
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<script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw1453","wgBackendResponseTime":152,"wgPageParseReport":{"limitreport":{"cputime":"2.164","walltime":"2.938","ppvisitednodes":{"value":14299,"limit":1000000},"postexpandincludesize":{"value":305854,"limit":2097152},"templateargumentsize":{"value":15389,"limit":2097152},"expansiondepth":{"value":16,"limit":100},"expensivefunctioncount":{"value":29,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":251823,"limit":5000000},"entityaccesscount":{"value":4,"limit":400},"timingprofile":["100.00% 1559.214 1 -total"," 27.66% 431.250 2 Template:Reflist"," 16.21% 252.708 35 Template:Cite_journal"," 10.95% 170.782 34 Template:Cite_book"," 8.70% 135.628 56 Template:Harvtxt"," 6.61% 103.041 2 Template:Blockquote"," 6.22% 97.008 1 Template:Cite_Q"," 4.88% 76.065 1 Template:Probability_fundamentals"," 4.80% 74.795 6 Template:Citation_needed"," 4.64% 72.419 1 Template:Sidebar"]},"scribunto":{"limitreport-timeusage":{"value":"0.935","limit":"10.000"},"limitreport-memusage":{"value":17735147,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREFAldrichMiller\"] = 2,\n [\"CITEREFAmariNagaoka2000\"] = 1,\n [\"CITEREFBarak2006\"] = 1,\n [\"CITEREFBasuLaha1954\"] = 1,\n [\"CITEREFBernardoSmith2000\"] = 1,\n [\"CITEREFBryc1995\"] = 1,\n [\"CITEREFCasellaBerger2001\"] = 1,\n [\"CITEREFCody1969\"] = 1,\n [\"CITEREFCoverThomas,_Joy_A.2006\"] = 1,\n [\"CITEREFCoverThomas2006\"] = 1,\n [\"CITEREFDas2021\"] = 1,\n [\"CITEREFDia2023\"] = 1,\n [\"CITEREFFan1991\"] = 1,\n [\"CITEREFGalambosSimonelli2004\"] = 1,\n [\"CITEREFGalton1889\"] = 1,\n [\"CITEREFGauss1809\"] = 1,\n [\"CITEREFGould1981\"] = 1,\n [\"CITEREFHalperinHartleyHoel1965\"] = 1,\n [\"CITEREFHart1968\"] = 1,\n [\"CITEREFHerrnsteinMurray1994\"] = 1,\n [\"CITEREFHuxley1932\"] = 1,\n [\"CITEREFJaynes2003\"] = 1,\n [\"CITEREFJohn1982\"] = 1,\n [\"CITEREFJohnsonKotzBalakrishnan1994\"] = 1,\n [\"CITEREFJohnsonKotzBalakrishnan1995\"] = 1,\n [\"CITEREFJordan2010\"] = 1,\n [\"CITEREFJorgeStephan2006\"] = 1,\n [\"CITEREFKarney2016\"] = 1,\n [\"CITEREFKindermanMonahan1977\"] = 1,\n [\"CITEREFKrishnamoorthy2006\"] = 1,\n [\"CITEREFKruskalStigler1997\"] = 1,\n [\"CITEREFLaplace1774\"] = 1,\n [\"CITEREFLaplace1812\"] = 1,\n [\"CITEREFLe_CamLo_Yang2000\"] = 1,\n [\"CITEREFLehmann1997\"] = 1,\n [\"CITEREFLeva1992\"] = 1,\n [\"CITEREFLexis1878\"] = 1,\n [\"CITEREFLukacs,_Eugene\"] = 1,\n [\"CITEREFLukacs1942\"] = 1,\n [\"CITEREFLukacsKing1954\"] = 1,\n [\"CITEREFMarsaglia2004\"] = 1,\n [\"CITEREFMarsagliaTsang2000\"] = 1,\n [\"CITEREFMaxwell1860\"] = 1,\n [\"CITEREFMcPherson1990\"] = 1,\n [\"CITEREFMonahan1985\"] = 1,\n [\"CITEREFNortonKhokhlovUryasev2019\"] = 1,\n [\"CITEREFOosterbaan1994\"] = 1,\n [\"CITEREFPapoulis\"] = 1,\n [\"CITEREFParkBera2009\"] = 1,\n [\"CITEREFPatelRead1996\"] = 1,\n [\"CITEREFPearson1901\"] = 1,\n [\"CITEREFPearson1905\"] = 1,\n [\"CITEREFPearson1920\"] = 1,\n [\"CITEREFQuine1993\"] = 1,\n [\"CITEREFRohrbasserVéron2003\"] = 1,\n [\"CITEREFScottNowak2003\"] = 1,\n [\"CITEREFShore1982\"] = 1,\n [\"CITEREFShore2005\"] = 1,\n [\"CITEREFShore2011\"] = 1,\n [\"CITEREFShore2012\"] = 1,\n [\"CITEREFSmith2000\"] = 1,\n [\"CITEREFStigler1978\"] = 1,\n [\"CITEREFStigler1982\"] = 1,\n [\"CITEREFStigler1986\"] = 1,\n [\"CITEREFStigler1999\"] = 1,\n [\"CITEREFSunKongPal2021\"] = 1,\n [\"CITEREFWalker1985\"] = 1,\n [\"CITEREFWallace1996\"] = 1,\n [\"CITEREFWeisstein\"] = 2,\n [\"CITEREFWest2009\"] = 1,\n [\"CITEREFWichura1988\"] = 1,\n [\"CITEREFWilliams2001\"] = 1,\n [\"CITEREFZelenSevero1964\"] = 1,\n [\"CITEREFde_Moivre1738\"] = 1,\n [\"Log-likelihood\"] = 1,\n}\ntemplate_list = table#1 {\n [\"!\"] = 2,\n [\"'\"] = 1,\n [\"=\"] = 5,\n [\"AS ref\"] = 1,\n [\"Abs\"] = 1,
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