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Multinomial Distribution

Overview

Multinomial distribution models the probability of each combination of successes in a series of independent trials. Use this distribution when there are more than two possible mutually exclusive outcomes for each trial, and each outcome has a fixed probability of success.

Parameter

Multinomial distribution uses the following parameter.

ParameterDescriptionConstraints
probabilitiesOutcome probabilities0probabilities(i)1;all(i)probabilities(i)=1

Probability Density Function

The multinomial pdf is

f(x|n,p)=n!x1!xk!p1x1pkxk,

where k is the number of possible mutually exclusive outcomes for each trial, and n is the total number of trials. The vector x = (x1...xk) is the number of observations of each k outcome, and contains nonnegative integer components that sum to n. The vector p = (p1...pk) is the fixed probability of each k outcome, and contains nonnegative scalar components that sum to 1.

Descriptive Statistics

The expected number of observations of outcome i in n trials is

E{xi}=npi,

where pi is the fixed probability of outcome i.

The variance is of outcome i is

var(xi)=npi(1pi).

The covariance of outcomes i and j is

cov(xi,xj)=npipj,ij.

Relationship to Other Distributions

The multinomial distribution is a generalization of the binomial distribution. While the binomial distribution gives the probability of the number of "successes" in n independent trials of a two-outcome process, the multinomial distribution gives the probability of each combination of outcomes in n independent trials of a k-outcome process. The probability of each outcome in any one trial is given by the fixed probabilities p1,..., pk.

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