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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.

Parameter | Description | Constraints |
---|---|---|

probabilities | Outcome probabilities |

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* = (*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.

The expected number of observations of outcome *i* in *n* trials
is

where *p _{i}* is
the fixed probability of outcome

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

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