Multinomial distribution

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polynomial distribution

The joint distribution of random variables that is defined for any set of non-negative integers satisfying the condition , , , by the formula


where (, ) are the parameters of the distribution. A multinomial distribution is a multivariate discrete distribution, namely the distribution for the random vector with (this distribution is in essence -dimensional, since it is degenerate in the Euclidean space of dimensions). A multinomial distribution is a natural generalization of a binomial distribution and coincides with the latter for . The name of the distribution is given because the probability (*) is the general term in the expansion of the multinomial . The multinomial distribution appears in the following probability scheme. Each of the random variables is the number of occurrences of one of the mutually exclusive events , , in repeated independent trials. If in each trial the probability of event is , , then the probability (*) is equal to the probability that in trials the events will appear times, respectively. Each of the random variables has a binomial distribution with mathematical expectation and variance .

The random vector has mathematical expectation and covariance matrix , where

(the rank of the matrix is because ). The characteristic function of a multinomial distribution is

For , the distribution of the vector with normalized components

tends to a certain multivariate normal distribution, while the distribution of the sum

(which is used in mathematical statistics to construct the "chi-squared" test) tends to the "chi-squared" distribution with degrees of freedom.


[1] H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946)



[a1] N.L. Johnson, S. Kotz, "Discrete distributions" , Wiley (1969)
How to Cite This Entry:
Multinomial distribution. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article