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In probability theory, the **multinomial distribution** is a generalization of the binomial distribution. The binomial distribution is the probability distribution of the number of "successes" in *n* independent Bernoulli trials, with the same probability of "success" on each trial. Instead of each trial resulting in "success" or "failure", imagine that each trial results in one of some fixed finite number *k* of possible outcomes, with probabilities *p*_{1}, ..., *p*_{k}, and there are *n* independent trials. In effect, it is the probability distribution of a random vector

satisfying the constraint

The probabilities are given by

for non-negative integers *x*_{1}, ..., *x*_{k}, if

and 0 otherwise.

Each of the *k* components separately has a binomial distribution with parameters *n* and *p*_{i}, for the appropriate value of the subscript *i*, and, because of the constraint that the sum of the components is *n*, they are negatively correlated.

The expected value is

The covariance matrix is as follows. Each diagonal entry is the variance of a binomially distributed random variable, and is therefore

The off-diagonal entries are the covariances. These are

for *i*, *j* distinct. This is a *k* × *k* nonnegative-definite matrix of rank *k* − 1.

The Dirichlet distribution is the conjugate prior of the multinomial in Bayesian statistics.

## See also

- Multinomial theorem

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