WalleniusHypergeometricDistribution

WalleniusHypergeometricDistribution[n,n_succ,n_tot,w]

represents a Wallenius noncentral hypergeometric distribution.

Details

A Wallenius hypergeometric distribution gives the distribution of the number of successes in n dependent draws from a population of size n_tot containing n_succ successes with the odds ratio w.
The probability for integer value in a Wallenius hypergeometric distribution is equal to $TemplateBox[{{n, _, {(, succ, )}}, x}, Binomial] TemplateBox[{{{n, _, {(, tot, )}}, -, {n, _, {(, succ, )}}}, {n, -, x}}, Binomial] int_0^1(1-t^(1/d))^(n-x) (1-t^(w/d))^xdt$ , where .
WalleniusHypergeometricDistribution allows n, n_succ, and n_tot to be any integers such that 0<n≤n_tot, 0≤n_succ≤n_tot, and w is any positive real number.
WalleniusHypergeometricDistribution can be used with such functions as Mean, CDF, and RandomVariate.

Background & Context

WalleniusHypergeometricDistribution[n,n_succ,n_tot,w] represents a discrete statistical distribution defined for integer values contained in and determined by four parameters n, n_succ, n_tot, and w. Here, w is a real number representing the odds ratio of the experiment described by the Wallenius hypergeometric distribution, while n, n_succ, and n_tot are integers satisfying 0<n≤n_tot and 0<n_succ≤n_tot, which represent the number of draws of the experiment, the number of successes within that population, and the size of the population drawn from, respectively. The Wallenius hypergeometric distribution has a probability density function (PDF) that is discrete and unimodal. The distribution is sometimes also referred to as Wallenius's noncentral hypergeometric distribution to differentiate it from the (central) hypergeometric distribution (HypergeometricDistribution in the Wolfram Language).
Wallenius's hypergeometric distribution arises in a particular urn model containing n_succ blue balls and n_tot-n_succ green balls having weights w₁ and w₂, respectively. Before the experiment, a fixed number n of balls are drawn at random, so that the probability of taking a particular ball is proportional to its weight and is dependent upon what happens to the other balls. Under this construction, the conditional distribution modeling the number of taken blue balls given n is modeled by Wallenius's hypergeometric distribution with w=w₁/w₂. (Note that this model is almost identical to the urn model defining FisherHypergeometricDistribution, with the exception that the latter is modeled by a drawing procedure that is independent, so that each individual draw is modeled by BinomialDistribution.)
A number of real-world phenomena can be modeled using a Wallenius hypergeometric distribution. For example, the distribution has been shown to model the deaths of species competing for a limited food resource (assuming the fates of the species members are dependent upon one another). Wallenius's hypergeometric distribution is also important to the theory of Monte Carlo simulations and is considered to be a generalized model of biased sampling.
RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a Wallenius hypergeometric distribution. Distributed[x,WalleniusHypergeometricDistribution[n,n_succ,n_tot,w]], written more concisely as xWalleniusHypergeometricDistribution[n,n_succ,n_tot,w], can be used to assert that a random variable x is distributed according to a Wallenius hypergeometric distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
The probability density and cumulative distribution functions may be given using PDF[WalleniusHypergeometricDistribution[n,n_succ,n_tot,w],x] and CDF[WalleniusHypergeometricDistribution[n,n_succ,n_tot,w],x], though one should note that there is no closed-form expression for its PDF. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively. These quantities can be visualized using DiscretePlot.
DistributionFitTest can be used to test if a given dataset is consistent with a Wallenius hypergeometric distribution, EstimatedDistribution to estimate a Wallenius hypergeometric parametric distribution from given data, and FindDistributionParameters to fit data to a Wallenius hypergeometric distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic Wallenius hypergeometric distribution and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic Wallenius hypergeometric distribution.
TransformedDistribution can be used to represent a transformed Wallenius hypergeometric distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain a Wallenius hypergeometric distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving Wallenius hypergeometric distributions.
WalleniusHypergeometricDistribution is related to a number of other statistical distributions. As mentioned previously, there is a fundamental link between WalleniusHypergeometricDistribution, FisherHypergeometricDistribution, and HypergeometricDistribution. The latter relationship can be made quantitatively precise by noting that FisherHypergeometricDistribution[n,n_succ,n_tot,1] has the same PDF as HypergeometricDistribution[n,n_succ,n_tot].

Examples

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Basic Examples (4)

Probability mass 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:

Scope (4)

Generate a sample of pseudorandom numbers from a noncentral hypergeometric distribution:

Compare its histogram to the PDF:

Distribution parameters estimation:

Estimate a distribution parameter 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: