A note on the distribution of the product of zero-mean correlated normal random variables

摘要

The problem of finding an explicit formula for the probability density function of two zero-mean correlated normal random variables dates back to 1936. Perhaps, surprisingly, this problem was not resolved until 2016. This is all the more surprising given that a very simple proof is available, which is the subject of this note; we identify the product of two zero-mean correlated normal random variables as a variance-gamma random variable, from which an explicit formula for the probability density function is immediate.