BUILT-IN MATHEMATICA SYMBOL

WalleniusHypergeometricDistribution

represents a Wallenius noncentral hypergeometric distribution.

MORE INFORMATION

A Wallenius hypergeometric distribution gives the distribution of the number of successes in n dependent draws from a population of size containing successes with the odds ratio w.

The probability for integer value in a Wallenius hypergeometric distribution is equal to where .

WalleniusHypergeometricDistribution allows n, , and to be any integers such that , , and w is any positive real number.

WalleniusHypergeometricDistribution can be used with such functions as Mean, CDF, and RandomVariate.

EXAMPLES

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Basic Examples (4)

Probability density function:

Cumulative distribution function:

Mean does not have closed form but can be obtained numerically:

Variance does not have closed form but can be obtained numerically:

Probability density function:

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Cumulative distribution function:

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Mean does not have closed form but can be obtained numerically:

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Variance does not have closed form but can be obtained numerically:

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Scope (4)

Generate a set of pseudorandom numbers that are hypergeometrically distributed:

Compare its histogram to the PDF:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Compare the density histogram of the sample with the PDF of the estimated distribution:

Hazard function:

Quantile function:

Applications (3)

The CDF of WalleniusHypergeometricDistribution is an example of a right-continuous function:

An urn contains

red balls of weight

and

blue balls of weight

. With

balls drawn one by one, the probability of drawing a particular ball is equal to the proportion of the total weight of all balls remaining in the urn at that moment. If

,

, and

, find the distribution of the number of red balls drawn:

Find the probability that at least 3 red balls were drawn:

Find the average number of red balls:

Simulate the number of red balls in 30 consecutive samples of 12:

Urn sampling leading to WalleniusHypergeometricDistribution can be simulated using RandomSample:

Properties & Relations (2)

Relationships to other distributions:

HypergeometricDistribution is a special case of Wallenius noncentral hypergeometric distribution: