RANDMULTINOMIAL Function :: SAS/IML(R) 13.1 User's Guide

RANDMULTINOMIAL Function

RANDMULTINOMIAL (N, NumTrials, Prob) ;

The RANDMULTINOMIAL function is part of the IMLMLIB library. The RANDMULTINOMIAL function generates a random sample from a multinomial distribution, which is a multivariate generalization of the binomial distribution.

The input parameters are as follows:

N: is the number of observations to sample.
NumTrials: is the number of trials. $NumTrials[j] \geq 0$ , for $j = 1 \ldots p$ .
Prob: is a $1 \times p$ vector of probabilities with $0 < \mi {Prob}[j] \leq 1$ and $\Sigma _{j=1}^ p \mbox{Prob}[j] = 1$ .

For each trial, $\mbox{Prob}[j]$ is the probability of event $E_ j$ , where the $E_ j$ are mutually exclusive and $\Sigma _{j=1}^ p \mbox{Prob}[j] = 1$ .

The RANDMULTINOMIAL function returns an $N \times p$ matrix that contains $N$ observations of $\mbox{NumTrials}$ random draws from the multinomial distribution. Each row of the resulting matrix is an integer vector $\{ X_1 \, X_2 \, \ldots \, X_ p\}$ with $\Sigma X_ j = \mbox{NumTrials}$ . That is, for each row, $X_ j$ indicates how many times event $E_ j$ occurred in $\mbox{NumTrials}$ trials.

If $X=\{ X_1 \, X_2 \, \ldots \, X_ p\}$ follows a multinomial distribution with $n$ trials and probabilities $\rho = \{ \rho _1 \, \rho _2 \, \ldots \, \rho _ p\}$ , then

the probability density function for $x$ is

$f(x; n, \rho )= \frac{n!}{ \prod _{i=1}^ p {x_ i!}} \prod _{i=1}^ p {{\rho _ i}^{x_ i}}$
the expected value of $X_ i$ is $n \rho _ i$ .
the variance of $X_ i$ is $n \rho _ i (1-\rho _ i)$ .
the covariance of $X_ i$ with $X_ j$ is $-n \rho _ i\rho _ j$ .
if $p=1$ then $X$ is constant.
if $p=2$ then $X_1$ is Binomial $(n, \rho _1)$ and $X_2$ is Binomial $(n, \rho _2)$ .

The following example generates 1,000 samples from a multinomial distribution with three mutually exclusive events. For each sample, 10 events are generated. Each row of the returned matrix x represents the number of times each event is observed. The example also computes the sample mean and covariance and compares them with the expected values.

call randseed(1);
prob = {0.3,0.6,0.1};
NumTrials = 10;
N = 1000;
x = RandMultinomial(N,NumTrials,prob);

/* population mean and covariance */
Mean = NumTrials * prob`;
Cov = -NumTrials*prob*prob`;
/* replace diagonal elements of Cov with Variance */
Variance = NumTrials*prob#(1-prob);
do i = 1 to nrow(prob);
   Cov[i,i] = Variance[i];
end;

SampleMean = mean(x);
SampleCov = cov(x);
print SampleMean Mean, SampleCov Cov;

Figure 24.300: Estimated Mean and Covariance Matrix

SampleMean			Mean
2.891	6.059	1.05	3	6	1

SampleCov			Cov
2.0051241	-1.69126	-0.313864	2.1	-1.8	-0.3
-1.69126	2.3198388	-0.628579	-1.8	2.4	-0.6
-0.313864	-0.628579	0.9424424	-0.3	-0.6	0.9

For further details about sampling from the multinomial distribution, see Gentle (2003), or Fishman (1996).