Language Reference

RANDNORMAL Function

RANDNORMAL (N, Mean, Cov );

The RANDNORMAL function is part of the IMLMLIB library . The RANDNORMAL function returns an $N \times p$ matrix that contains N random draws from the multivariate normal distribution with mean vector Mean and covariance matrix Cov.

The inputs are as follows:

N

is the number of desired observations sampled from the multivariate normal distribution.

Mean

is a $1 \times p$ vector of means.

Cov

is a $p \times p$ symmetric positive definite variance-covariance matrix.

If X follows a multivariate normal distribution with mean vector $\mu $ and variance-covariance matrix $\Sigma $, then

  • the probability density function for x is

    \[ f(x; \mu , \Sigma ) = \frac{1}{(2 \pi )^{p/2} |\Sigma |^{1/2}} \exp \left( - \frac{(x-\mu ) \Sigma ^{-1} (x-\mu )^ T}{2} \right) \]

  • if $p=1$, the probability density function reduces to a univariate normal distribution.

  • the expected value of $X_ i$ is $\mu _ i$.

  • the covariance of $X_ i$ and $X_ j$ is $\Sigma _{ij}$.

The following example generates 1,000 samples from a two-dimensional multivariate normal distribution with mean vector $(1 \, 2)$ and a given covariance matrix. Each row of the returned matrix x is a row vector sampled from the multivariate normal distribution. The example computes the sample mean and covariance and compares them with the expected values. 

call randseed(1);
N = 1000;
Mean = {1 2};
Cov = {2.4 3, 3 8.1};

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

Figure 25.307: Estimated Mean and Covariance Matrix

SampleMean   Mean  
1.0619604 2.1156084 1 2

SampleCov   Cov  
2.5513518 3.2729559 2.4 3
3.2729559 8.7099585 3 8.1 




            

            


            
For further details about sampling from the multivariate normal distribution, see Gentle (2003).