## 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
`probabilities`Outcome probabilities$0\le \text{probabilities}\left(i\right)\le 1\text{\hspace{0.17em}};\text{\hspace{0.17em}}\sum _{\text{all}\left(i\right)}\text{probabilities}\left(i\right)=1$

### Probability Density Function

The multinomial pdf is

`$f\left(x|n,p\right)=\frac{n!}{{x}_{1}!\cdots {x}_{k}!}{p}_{1}^{{x}_{1}}\cdots {p}_{k}^{{x}_{k}},$`

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

`$\text{E}\left\{{x}_{i}\right\}=n{p}_{i}\text{\hspace{0.17em}},$`

where pi is the fixed probability of outcome i.

The variance is of outcome i is

`$\text{var}\left({x}_{i}\right)=n{p}_{i}\left(1-{p}_{i}\right)\text{\hspace{0.17em}}.$`

The covariance of outcomes i and j is

`$\mathrm{cov}\left({x}_{i},{x}_{j}\right)=-n{p}_{i}{p}_{j}\text{\hspace{0.17em}},\text{\hspace{0.17em}}i\ne j.$`

### 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.