Why is the Multivariate Normal Distribution the only Multivariate Distribution we hear about

Question

The multivariate normal distribution is the only multivariate distribution we ever hear about. We never hear about other multivariate distributions such as the multivariate binomial, multivariate exponential, etc.

Is there any reason for this? Is this simply due to the fact that if multiple random variables individually have some distribution - it does not mean that their joint distribution has that same distribution at the multivariate level? Is it possible to prove this?

Thanks!

Answer 1

First, it's not and lots of places talk about other distributions, although it is more popular. But reasons for it's prominence could be:

• The multivariate CLT which is an extension of the CLT to higher dimensions: if we have a collection of i.i.d. random vectors $X_1, \ldots, X_n$ and same mean vector $\bar X$, then $\bar X$ can be approximated by a multivariate normal distribution.
• With eigen-decomposition we can turn and multivariate normal into a standard multivariate normal pretty easily, making it even simpler, which we can't do for other distributions
• Modeling, the same way that other multivariate distributions can model phenomenon, the multivariate can as well, it's just the multivariate normal has other uses which makes it more popular.

Also, it's just easier to work with then other distributions.