Multinomial Distribution

The multinomial distribution is a discrete probability density function.

Description

Where the binomial distribution requires two outcomes for any trial, the multinomial distribution allows for multiple exclusive and exhaustive outcomes.

There are k possible outcomes, each with an associated probability of p_i for outcome i. The count of observations that realize into outcome i is a random variable, denoted X_i. The probability of any particular outcome is expressed as Prob(X₁=x₁, X₂=x₂, ... X_k=x_k). Further note that it is required for...

• the sum of p_i across all outcomes is 1.

• the sum of x_i across all outcomes is n.

Moments

It doesn't necessarily make sense to consider the expected value of all multinomially distributed random variables at the same time.

For any one of the X_i random variables, the expected value is given as E[X_i] = np_i.

Variance is given as Var[X_i] = np_i(1 - p_i).

These of course match the binomial distribution.

The covariance matrix of a multinomial distribution is constructed as:

• variances on the diagonal, as above

• covariances calculated as Cov(X_i, X_j) = -np_ip_j

CategoryRicottone