39 lines
1.3 KiB
Markdown
39 lines
1.3 KiB
Markdown
# Property tests for squiggle
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Here are some property tests for squiggle. I am testing mostly for the mean and the standard deviation. I know that squiggle doesn't yet have functions for the standard deviation, but they could be added.
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The keywords to search for are "[algebra of random variables](https://wikiless.org/wiki/Algebra_of_random_variables?lang=en)".
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## Sums
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$$ mean(f+g) = mean(f) + mean(g) $$
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$$ Std(f+g) = sqrt(std(f)^2 + std(g)^2) $$
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In the case of normal distributions,
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$$ normal(a,b) + normal(c,d) = normal(a+c, sqrt(b^2 + d^2) $$
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## Substractions
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$$ mean(f-g) = mean(f) - mean(g) $$
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$$ std(f-g) = sqrt(std(f)^2 + std(g)^2) $$
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## Multiplications
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$$ mean(f \cdot g) = mean(f) \cdot mean(g) $$
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$$ std(f \cdot g) = sqrt( (std(f)^2 + mean(f)) \cdot (std(g)^2 + mean(g)) - (mean(f) \cdot mean(y))^2) $$
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## Divisions
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Divisions are tricky, and in general we don't have good expressions to characterize properties of ratios. In particular, the ratio of two normals is a Cauchy distribution, which doesn't have to have a mean.
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## To do:
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- Provide sources or derivations, useful as this document becomes more complicated
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- Provide definitions for the probability density function, exponential, inverse, log, etc.
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- Provide at least some tests for division
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- See if playing around with characteristic functions turns out anything useful
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