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

### 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* =
(*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}.