What is the distribution of a random variable that is the product of the two normal random variables ? 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...

For the special case that both Gaussian random variables $X$ and $Y$ have zero mean and unit variance, and are independent, the answer is that $Z=XY$ has the probability density $p_Z(z)={\rm K}_0(|z|)/\pi$. The brute force way to do this is via the transformation theorem:
\begin{align}
p_Z(z)&=\frac{1}{2\pi}\int_{-\infty}^\infty{\rm d}x\int_{-\infty}^\infty{\rm d}y\;{\rm e}^{-(x^2+y^2)/2}\delta(z-xy) \\
&= \frac{1}{\pi}\int_0^\infty\frac{{\rm d}x}{x}{\rm e}^{-(x^2+z^2/x^2)/2}\\
&= \frac{1}{\pi}{\rm K}_0(|z|) \ .
\end{align}
