Variance for a product-normal distribution I have two normally distributed random variables (zero mean), and I am interested in the distribution of their product; a normal product distribution. 
It's a strange distribution involving a delta function.
What is the variance of this distribution - and is it finite?
I know that
$Var(XY)=Var(X)Var(Y)+Var(X)E(Y)^2+Var(Y)E(X)^2$
However I'm running a few simulations and noticing that the sample average of variables following this distribution is not converging to normality - making me guess that its variance is not actually finite.

Hint: We need to know something about the joint distribution. The simplest assumption is that $X$ and $Y$ are independent. Let $W=XY$. We want $E(W^2)-(E(W))^2$. To calculate $E((XY)^2)$, use independence. 
