Multinomial distribution

polynomial distribution

2020 Mathematics Subject Classification: Primary: 60E99 [MSN][ZBL]

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.

References

[C]	H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946) MR0016588 Zbl 0063.01014

References

[JK]	N.L. Johnson, S. Kotz, "Discrete distributions" , Wiley (1969) MR0268996 Zbl 0292.62009