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, …, A_m have probabilities p₁, …, p_m, respectively, where 0 ≤ P_k < 1; k = 1, …, m; and p₁ + ··· + p_m = 1. Then the joint distribution of the random variables X₁, …, x_m, where X_k is the number of occurrences of the event A_k 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, …, n_m that satisfy the single condition n₁ + ··· + 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_k and variance np_k(1 – P_k). 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.