November 2012
Intermediate to advanced
794 pages
22h 9m
English
book
Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications
by John J. Shynk
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4.7 GENERATING DEPENDENT RANDOM VARIABLES
One can easily generate a multivariate pdf 
for N independent marginal pdfs as the product 
, and this can be done even if the marginal pdfs are from different distribution families as illustrated in previous examples. Unfortunately, there are not many widely used multivariate distributions for dependent random variables, as in the case of the bivariate Gaussian and Student's t distributions where the correlation coefficient ρ appears explicitly. Dependency can be introduced for other random variables by implementing the following procedure, which we describe for the bivariate case (N = 2). This method does not yield a closed-form expression for the desired joint pdf, but instead is a means of generating samples for a computer simulation:
- Let
be a random vector with a bivariate standard Gaussian pdf and a prespecified value for the correlation coefficient 
. Note that for the standard bivariate pdf, the mean is zero and 
. This matrix was defined ...
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