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)

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.