# Multinomial Distribution

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## multinomial distribution

[¦məl·tə¦nō·mē·əl ‚di·strə′byü·shən]## Multinomial Distribution

a joint probability distribution of random variables, each of which expresses the number of times one of several mutually exclusive events occurs in repeated independent trials. Suppose that in each trial the events *A*_{1}, …, *A*_{m} have probabilities *p*_{1}, …, *p _{m}*, respectively, where 0 ≤

*P*< 1;

_{k}*k*= 1, …,

*m*; and

*p*

_{1}+ ··· +

*p*= 1. Then the joint distribution of the random variables

_{m}*X*

_{1}, …,

*x*, where

_{m}*X*is the number of occurrences of the event

_{k}*A*in

_{k}*n*trials, is given by the probabilities

that is, the probability that the event *A*_{1} occurs *n*_{1} times in *n* independent trials, that the event *A*_{2} occurs *n*_{2} times, and so on. These probabilities are defined for every set of nonnegative integers *n*_{1}, …, *n*_{m} that satisfy the single condition *n*_{1} + ··· + *n _{m}* =

*n*.

The multinomial distribution is a natural generalization of the binomial distribution, to which it reduces when *m* = 2. Every random variable *X _{k}* in this case must have a binomial distribution with mathematical expectation

*np*and variance

_{k}*np*(1 –

_{k}*P*). As

_{k}*n*→ ∞, the joint distribution of the random variables

tends to some limiting normal distribution, and the sum

approaches a chi-square distribution with *n* – 1 degrees of freedom. This sum is used in mathematical statistics in the chi-square test.

### REFERENCES

Cramer, H.*Matematicheskie metody statistiki*. Moscow, 1948. (Translated from English.)

Feller, W.

*Vvedenie v teoriiu veroiatnostei i ee prilozheniia*, 2nd ed., vols. 1–2. Moscow, 1967. (Translated from English.)

A. V. PROKHOROV