# Multinomial Distribution

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## multinomial distribution

[¦məl·tə¦nō·mē·əl ‚di·strə′byü·shən]
(mathematics)
The joint distribution of the set of random variables which are the number of occurrences of the possible outcomes in a sequence of multinomial trials.

## 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 A1, …, Am have probabilities p1, …, pm, respectively, where 0 ≤ Pk < 1; k = 1, …, m; and p1 + ··· + pm = 1. Then the joint distribution of the random variables X1, …, xm, where Xk is the number of occurrences of the event Ak in n trials, is given by the probabilities

that is, the probability that the event A1 occurs n1 times in n independent trials, that the event A2 occurs n2 times, and so on. These probabilities are defined for every set of nonnegative integers n1, …, nm that satisfy the single condition n1 + ··· + nm = n.

The multinomial distribution is a natural generalization of the binomial distribution, to which it reduces when m = 2. Every random variable Xk in this case must have a binomial distribution with mathematical expectation npk and variance npk(1 – Pk). As 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

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