Multinomial distribution

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Probability mass function
Cumulative distribution function
Parameters number of trials (integer)
event probabilities ()
Probability mass function (pmf)
Cumulative distribution function (cdf)
Excess kurtosis
Moment-generating function (mgf)
Characteristic function

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, ..., pk (so that pi ≥ 0 for i = 1, ..., k and ), and there are n independent trials. Then let the random variables indicate the number of times outcome number i was observed over the n trials. follows a multinomial distribution with parameters n and p.


Probability mass function

The probability mass function of the multinomial distribution is:

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


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 pi, for the appropriate value of the subscript i.

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

the number of n-combinations of a multiset with k types, or multiset coefficient.

Related distributions

See also

External links


Evans, Merran (2000). Statistical Distributions. New York: Wiley. pp. 134–136. ISBN 0-471-37124-6. 3rd ed. Unknown parameter |coauthors= ignored (help)

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