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

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 = (x₁...x_k) is the number of observations of each k outcome, and contains nonnegative integer components that sum to n. The vector p = (p₁...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

$$\text{E}\left\{x_i\right\}=n{p}_i,$$

where p_i is the fixed probability of outcome i.

The variance is of outcome i is

$$\text{var}\left(x_i\right)=n{p}_i\left(1-p_i\right).$$

The covariance of outcomes i and j is

$$\mathrm{cov}(x_i,x_j)=-n{p}_i{p}_j,i\ne j.$$

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,..., p_k.