85 lines
2.1 KiB
JavaScript
85 lines
2.1 KiB
JavaScript
/**
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* @license Apache-2.0
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*
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* Copyright (c) 2018 The Stdlib Authors.
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*
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* Licensed under the Apache License, Version 2.0 (the "License");
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* you may not use this file except in compliance with the License.
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* You may obtain a copy of the License at
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*
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* http://www.apache.org/licenses/LICENSE-2.0
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*
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* Unless required by applicable law or agreed to in writing, software
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* distributed under the License is distributed on an "AS IS" BASIS,
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* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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* See the License for the specific language governing permissions and
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* limitations under the License.
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*/
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'use strict';
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// MODULES //
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var floor = require( '@stdlib/math/base/special/floor' );
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var ln = require( '@stdlib/math/base/special/ln' );
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// MAIN //
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/**
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* Returns a pseudorandom number drawn from a geometric distribution.
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*
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* ## Proof
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*
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* Consider
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*
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* ```tex
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* N = \left \lfloor \ln (U) / \ln (1-p) \right \rfloor
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* ```
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*
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* where \\( U \\) is uniform on the interval \\((0,1)\\). Accordingly, \\(N\\) must be a nonnegative integer, and, for every \\( n \geq 0\\), the event \\(A_n = \left \{ N = n \right \}\\) is
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*
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* ```tex
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* A_n = \left \{(n+1) \ln (1-p) < \ln (U) \leq n \ln (1-p) \right \}
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* ```
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*
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* where \\(\ln (1-p) < 0\\). Thus,
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*
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* ```tex
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* A_n = \left \{(1-p)^{n+1} < U \leq (1-p)^n \right \}
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* ```
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*
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* For every \\(u < v\\) on the interval \\((0,1)\\),
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*
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* ```tex
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* P\left \[u < U \leq v\right \] = v - u
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* ```
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*
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* Hence,
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*
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* ```tex
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* P\left \[N = n \right \] = P\left \[A_n\right \] = (1-p)^n - (1-p)^{n+1} = (1-p)^n(1-(1-p)) = p(1-p)^n
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* ```
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*
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* which proves that \\(N\\) is a geometric random variable.
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*
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*
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* @private
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* @param {PRNG} rand - PRNG for uniformly distributed numbers
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* @param {Probability} p - success probability
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* @returns {NonNegativeInteger} pseudorandom number
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*/
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function geometric( rand, p ) {
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var u = rand();
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if ( u === 0.0 ) {
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// Drawing random variates from a PRNG (with period > 1) is effectively sampling without replacement. Thus, should not be possible to draw `0` twice in a row.
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u = rand();
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}
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return floor( ln( u ) / ln( 1.0-p ) );
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}
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// EXPORTS //
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module.exports = geometric;
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