You can express a Student-t distribution as a continuous mixture of normal distributions. Some properties of the t distribution are easier to prove in this form. Here are notes with details.

I ran across this tidbit reading Bayesian Data Analysis by Gelman *et al*.

**Related post**: Beer, Wine, and Statistics (origin of the Student-t distribution)

I’m doing some research on mixtures, and I was surprised to see that Gelman (in that book, which I will one day find a cheap copy of) does a much better job of explaining the EM algorithm than any of my mixture references.

And it was hands down the best Bayesian book I looked at when trying to do a readings course on the topic last semester.

I think I’m trying to say: Yay Gelman!