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Multinomial Distribution

Let a set of random variates X_1, X_2, ..., X_n have a probability function

P(X_1=x_1,...,X_n=x_n)=(N!)/(product_(i=1)^(n)x_i!)product_(i=1)^ntheta_i^(x_i)
(1)

where x_i are nonnegative integers such that

sum_(i=1)^nx_i=N,
(2)

and theta_i are constants with theta_i>0 and

sum_(i=1)^ntheta_i=1.
(3)

Then the joint distribution of X_1, ..., X_n is a multinomial distribution and P(X_1=x_1,...,X_n=x_n) is given by the corresponding coefficient of the multinomial series

(theta_1+theta_2+...+theta_n)^N.
(4)

In the words, if X_1, X_2, ..., X_n are mutually exclusive events with P(X_1=x_1)=theta_1, ..., P(X_n=x_n)=theta_n. Then the probability that X_1 occurs x_1 times, ..., X_n occurs x_n times is given by

P_N(x_1,x_2,...,x_n)=(N!)/(x_1!...x_n!)theta_1^(x_1)...theta_n^(x_n).
(5)

(Papoulis 1984, p. 75).

The mean and variance of X_i are

mu_i=Ntheta_i
(6)
sigma_i^2=Ntheta_i(1-theta_i).
(7)

The covariance of X_i and X_j is

sigma_(ij)^2=-Ntheta_itheta_j.
(8)

See also

Binomial Distribution, Multinomial Coefficient

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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 Resource. https://mathworld.wolfram.com/MultinomialDistribution.html

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