MultivariateHypergeometricDistribution

MultivariateHypergeometricDistribution[n,{m1,m2,,mk}]
represents a multivariate hypergeometric distribution with n draws without replacement from a collection containing mi objects of type i.

DetailsDetails

  • The probability for a vector of non-negative integers , , , in a multinomial distribution is proportional to product_(i=1)^kTemplateBox[{{m, _, i}, {x, _, i}}, Binomial] given that .
  • The numbers mi can be any non-negative integers and n any positive integer less than or equal to m1++mk.
  • The number of trials n can be any positive integer and mi any non-negative integer.
  • MultivariateHypergeometricDistribution can be used with such functions as Mean, CDF, and RandomVariate.

Background & Context
Background & Context

  • MultivariateHypergeometricDistribution[n,{m1,m2,,mk}] represents a discrete multivariate statistical distribution supported over the subset of consisting of all tuples of integers satisfying and and characterized by the property that each of the ^(th) (univariate) marginal distributions is a HypergeometricDistribution for . In other words, each of the variables satisfies xjHypergeometricDistribution[n,mj,m1++mk] for . The multivariate hypergeometric distribution is parametrized by a positive integer n and by a vector {m1,m2,,mk} of non-negative integers that together define the associated mean, variance, and covariance of the distribution.
  • The multivariate hypergeometric distribution models a scenario in which n draws are made without replacement from a collection containing mi objects of type i. This can be visualized as an urn model in which n balls are drawn without replacement from an urn containing k different types of balls, with the condition that there are mi balls of type i for . The multivariate hypergeometric distribution was first analyzed in a 1708 essay by French mathematician Pierre Raymond de Montmort, making it one of the earliest studied multivariate probability distributions. It has since become a tool in the study of a number of different phenomena, including faulty inspection procedures, and is a widely utilized model in fields such as statistical decision theory.
  • RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a multivariate hypergeometric distribution. Distributed[x,MultivariateHypergeometricDistribution[n,{m1,m2,,mk}]] , written more concisely as xMultivariateHypergeometricDistribution[n,{m1,m2,,mk}], can be used to assert that a random variable x is distributed according to a multivariate 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 for multivariate hypergeometric distributions may be given using PDF[MultivariateHypergeometricDistribution[n,{m1,m2,,mk}]] and CDF[MultivariateHypergeometricDistribution[n,{m1,m2,,mk}]]. The mean, median, variance, covariance, raw moments, and central moments may be computed using Mean, Median, Variance, Covariance, Moment, and CentralMoment, respectively.
  • DistributionFitTest can be used to test if a given dataset is consistent with a multivariate hypergeometric distribution, EstimatedDistribution to estimate a multivariate hypergeometric parametric distribution from given data, and FindDistributionParameters to fit data to a multivariate hypergeometric distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic multivariate hypergeometric distribution, and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic multivariate hypergeometric distribution.
  • TransformedDistribution can be used to represent a transformed multivariate 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 multivariate hypergeometric distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving multivariate hypergeometric distributions.
  • MultivariateHypergeometricDistribution is related to a number of other distributions. It is connected to HypergeometricDistribution as discussed above, and while the one-dimensional marginal PDFs of MultivariateHypergeometricDistribution are each a HypergeometricDistribution, the multivariate marginals do not simplify to named distributions. The urn model for MultivariateHypergeometricDistribution is related to that of MultinomialDistribution, in the sense that the latter distribution models drawing with replacement. Because of its relation to the univariate HypergeometricDistribution, MultivariateHypergeometricDistribution is also related to GeometricDistribution, NormalDistribution, PoissonDistribution, PearsonDistribution, and BetaBinomialDistribution.

ExamplesExamplesopen allclose all

Basic Examples  (4)Basic Examples  (4)

Probability mass function:

In[1]:=
Click for copyable input
Out[1]=
In[2]:=
Click for copyable input
Out[2]=

Cumulative distribution function:

In[1]:=
Click for copyable input
Out[1]=

Mean and variance:

In[1]:=
Click for copyable input
Out[1]=
In[2]:=
Click for copyable input
Out[2]=

Covariance:

In[1]:=
Click for copyable input
Out[1]//MatrixForm=
Introduced in 2010
(8.0)