## 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.

Parameter | Description | Constraints |

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

### 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* = (*x*_{1}...*x*_{k})
is the number of observations of each *k* outcome,
and contains nonnegative integer components that sum to *n*.
The vector *p* = (*p*_{1}...*p*_{k})
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 *p*_{i} 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

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

### 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 *p*_{1},..., *p*_{k}.

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