Multinomial distribution

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Multinomial
Probability mass function
Cumulative distribution function
Parameters number of trials (integer)
event probabilities ()
Support
Probability mass function (pmf)
Cumulative distribution function (cdf)
Mean
Median
Mode
Variance
Skewness
Excess kurtosis
Entropy
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.

Specification

Probability mass function

The probability mass function of the multinomial distribution is:

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

Properties

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

References

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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