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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. In a multinomial distribution, each trial results in exactly one of some fixed finite number k of possible outcomes, with probabilities p₁, ..., p_k (so that p_i ≥ 0 for i = 1, ..., k and ), and there are n independent trials. Then let the random variables X_i indicates the number of times outcome number i was observed over the n trials. follows a multinomial distribution with parameters n and p.

Contents 1 Specification 1.1 Probability mass function 2 Properties 3 Related distributions 4 See also

The probability mass function of the multinomial distribution is:

for non-negative integers x₁, ..., x_k.

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:

for i, j distinct.

All covariances are negative because for fixed N, an increase in one component of a multinomial vector requires a decrease in another component.

This is a k × k nonnegative-definite matrix of rank k − 1.

The off-diagonal entries of the corresponding correlation matrix are

Note that the sample size drops out of this expression.

Each of the k components separately has a binomial distribution with parameters n and p_i, for the appropriate value of the subscript i.

The support of the multinomial distribution is the set : Its number of elements is :

• When k=2, the multinomial distribution is the binomial distribution.
• The Dirichlet distribution is the conjugate prior of the multinomial in Bayesian statistics.

• Multinomial theorem

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