TITLE: What is the probability distribution function for the product of two correlated Gaussian random variable?
QUESTION [11 upvotes]: Suppose we have pair $(X,Y)\sim Normal([\mu_x,\mu_y],{{\sigma_x^2\atop\rho \sigma_x\sigma_y } {\rho \sigma_x\sigma_y \atop \sigma_y^2} }] $
How is $U=X\cdot Y$ distributed?
I've tried to compute this by substituting y=u/x  in the bivariate normal pdf and taking integral(from $-\infty$ to $\infty$ with respect to x. I find the pdf of U as the sum of two  exponential distributions(one for U<0 and one for U>0) that are weighted unequally. Is my method valid, or do I have to deal with cumulative distribution functions instead?

REPLY [4 votes]: LaGatta's answer nails it, and may be useful for drawing simulations, etc.
This is just a note to remind that if one is only interested in the mean of the product of normally-distributed (possibly correlated) random variables, then the answer is straightforward, using the identity $\operatorname{E}XY= \operatorname{Cov}(X,Y) + (\operatorname{E}X)(\operatorname{E}Y)$.