What is the distribution of a random variable that is the product of the two normal random variables ?
Let $X\sim N(\mu_1,\sigma_1), Y\sim N(\mu_2,\sigma_2)$ and $Z=XY$
That is, what is its probability density function, its expected value, and its variance ?
I'm kind of stuck and I can't find a satisfying answer on the web. If anybody knows the answer, or a reference or link, I would be really thankful...