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.
Multinomial distribution uses the following parameter.
The multinomial pdf is
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.
The expected number of observations of outcome i in n trials is
where pi is the fixed probability of outcome i.
The variance is of outcome i is
The covariance of outcomes i and j is
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.