TOPICS

# Multinomial Distribution

Let a set of random variates , , ..., have a probability function

 (1)

where are nonnegative integers such that

 (2)

and are constants with and

 (3)

Then the joint distribution of , ..., is a multinomial distribution and is given by the corresponding coefficient of the multinomial series

 (4)

In the words, if , , ..., are mutually exclusive events with , ..., . Then the probability that occurs times, ..., occurs times is given by

 (5)

(Papoulis 1984, p. 75).

The mean and variance of are

 (6) (7)

The covariance of and is

 (8)

## See also

Binomial Distribution, Multinomial Coefficient

## Explore with Wolfram|Alpha

More things to try:

## References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 532, 1987.Papoulis, A. Probability, Random Variables, and Stochastic Processes, 2nd ed. New York: McGraw-Hill, 1984.

## Referenced on Wolfram|Alpha

Multinomial Distribution

## Cite this as:

Weisstein, Eric W. "Multinomial Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MultinomialDistribution.html