# 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.

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The characteristic function of this distribution is $f(z)=\dfrac{1}{\sqrt{1+t^2}}$, you can find the mean and variance from this. (If they exist) – Inquest Feb 11 '13 at 0:16
How do I do that? Sorry - I am not that familiar with characteristic functions. – s4027340 Feb 11 '13 at 0:43

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.
Yes. And without replacing the assumption of independence by some other explicit assumption, we cannot compute the variance of $XY$. – André Nicolas Feb 11 '13 at 5:02