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...

To your first point, let's look at how we calculate the expectation of a product. The idea is easy enough but Gaussian distributions can look a little messier than they really are.
Specifically, to look at the distribution function, I would start here: http://en.wikipedia.org/wiki/Product_distribution
For expectation though, recall (or note):
For independent random variables, the joint probability distribution function, $h(x,y)$ can be found simply as the product of the marginal distributions, say $f(x)$ and $g(y)$.
That is $h(x,y)=f(x)*g(y)$. You find the expectation in the same way you would find it for a single variable with single pmf. Namely,
$E(XY)=E(Z)=\int\int xy*h(x,y)dydx$
=$\int\int xy (f(x)g(y)dydx)=[\int xf(x)dx][\int yg(y)dy]=E(X)E(Y)$
For standard normal RVs, this is simple to compute. If, in fact, your variables are not independent, then you need to incorporate a covariance term into your calculations.
Hope that helps.
