Formula for the variance of the product of two random variables Let's say I have two random variables $X$ and $Y$.  Can I write that:
$$VAR \left[XY\right] = \left(E\left[X\right]\right)^2 VAR \left[Y\right] + \left(E\left[Y\right]\right)^2 VAR \left[X\right] + 2 \left(E\left[X\right]\right) \left(E\left[Y\right]\right) COV\left[X,Y\right]?$$
If this is not correct, how can I intuitively prove that?

COUNTEREXAMPLE
Let $X=Y\sim N(0,1)$.
Since both have expected value zero, the right-hand side is zero.
However, $XY\sim\chi^2_1$, which has a variance of $2$.
