Multinomial Distribution
Let a set of random variates X_1, X_2, ..., X_n have a probability function
where x_i are nonnegative integers such that
| [画像: sum_(i=1)^nx_i=N, ] |
(2)
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and theta_i are constants with theta_i>0 and
| [画像: sum_(i=1)^ntheta_i=1. ] |
(3)
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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)
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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
(Papoulis 1984, p. 75).
The mean and variance of X_i are
The covariance of X_i and X_j is
| sigma_(ij)^2=-Ntheta_itheta_j. |
(8)
|
See also
Binomial Distribution, Multinomial CoefficientExplore with Wolfram|Alpha
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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 DistributionCite this as:
Weisstein, Eric W. "Multinomial Distribution." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MultinomialDistribution.html