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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, \ldots, X_n=x_n)={N!\over\prod_{i=1}^n x_i!} \prod_{i=1}^n{\theta_i}^{x_i}
\end{displaymath} (1)

where $x_i$ are Positive Integers, $\theta_i>0$, and
\sum_{i=1}^n \theta_i=1
\end{displaymath} (2)

\sum_{i=1}^n x_i=N.
\end{displaymath} (3)

Then the joint distribution of $X_1$, ..., $X_n$ is a multinomial distribution and $P(X_1=x_1, \ldots, X_n=x_n)$ is given by the corresponding coefficient of the Multinomial Series
\end{displaymath} (4)

The Mean and Variance of $X_i$ are
$\displaystyle \mu_i$ $\textstyle =$ $\displaystyle N\theta_i$ (5)
$\displaystyle {\sigma_i}^2$ $\textstyle =$ $\displaystyle N\theta_i(1-\theta_i).$ (6)

The Covariance of $X_i$ and $X_j$ is
\end{displaymath} (7)

See also Binomial Distribution


Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 532, 1987.

© 1996-9 Eric W. Weisstein