### Multivariate Random Variables

mathStatica extends naturally to a multivariate setting. To illustrate, let us suppose that and have joint pdf with support , :

where parameter α is such that . This is known as a Gumbel bivariate Exponential distribution. Here is a plot of :

Fig. 1: A Gumbel bivariate Exponential pdf when

Here is the cdf, namely :

Here is , the covariance between and :

More generally, here is the variance-covariance matrix:

Here is the marginal pdf of :

Here is the conditional pdf of , given :

Here is the bivariate mgf :

Differentiating the mgf is one way to derive moments. Here is the product moment :

which we could otherwise have found directly with:

Multivariate transformations pose no problem to mathStatica either. For instance, let and denote transformations of and . Then our transformation equation is:

Using Transform, we can find the joint pdf of random variables and , denoted :

while the extremum of the domain of support of the new random variables are: