# MultinomialDistribution

MultinomialDistribution[n,{p1,p2,,pm}]

represents a multinomial distribution with n trials and probabilities pi.

# Details

• The probability for a vector of non-negative integers , , , in a multinomial distribution is .
• The number of trials n can be any positive integer, and can be any non-negative real numbers such that .
• MultinomialDistribution allows n and pi to be dimensionless quantities. »
• MultinomialDistribution can be used with such functions as Mean, CDF, and RandomVariate.

# Background & Context

• MultinomialDistribution[n,{p1,p2,,pm}] 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 (univariate) marginal distributions has a BinomialDistribution for . In other words, each of the variables satisfies xjBinomialDistribution[n,pj] for . The multinomial distribution is parametrized by a positive integer n and a vector {p1,p2,,pm} of non-negative real numbers satisfying , which together define the associated mean, variance, and covariance of the distribution.
• The multinomial distribution models a scenario in which n draws are made with replacement from a collection with type i representing pi percent of the total population. This can be visualized as an urn model in which n balls are drawn with replacement from an urn containing m different types of balls with the condition that the percentage of balls of type i is pi balls for . The multinomial 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 kinetic theory in particle physics and overstatement in accounting. It is also widely used in the analysis of contingency tables, telecommunications modeling, medical epidemiology, and photon counting.
• RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a multinomial distribution. Distributed[x,MultinomialDistribution[n,{p1,p2,,pm}]] , written more concisely as xMultinomialDistribution[n,{p1,p2,,pm}], can be used to assert that a random variable x is distributed according to a multinomial 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 multinomial distributions may be given using PDF[MultinomialDistribution[n,{p1,p2,,pm}]] and CDF[MultinomialDistribution[n,{p1,p2,,pm}]]. 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 multinomial distribution, EstimatedDistribution to estimate a multinomial parametric distribution from given data, and FindDistributionParameters to fit data to a multinomial distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic multinomial distribution, and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic multinomial distribution.
• TransformedDistribution can be used to represent a transformed multinomial 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 multinomial distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving multinomial distributions.
• MultinomialDistribution is related to a number of other distributions. MultinomialDistribution is connected to the BinomialDistribution as discussed above and, while the one-dimensional marginal PDFs of the MultinomialDistribution each have a BinomialDistribution, the multivariate marginals do not simplify to named distributions. The urn model for MultinomialDistribution is related to that of MultivariateHypergeometricDistribution in the sense that the latter distribution models drawing without replacement. Because of its relation to the univariate BinomialDistribution, it is also related to BernoulliDistribution, NormalDistribution, PoissonDistribution, BetaBinomialDistribution, and NegativeBinomialDistribution.

# Examples

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## Basic Examples(4)

Probability mass function:

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Cumulative distribution function:

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Mean and variance:

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Covariance:

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