Is the product of two Gaussian random variables also a Gaussian?

Question 1

Say I have $X \sim \mathcal N(a, b)$ and $Y\sim \mathcal N(c, d)$ . Is $XY$ also normally distributed?

Is the answer any different if we know that $X$ and $Y$ are independent?

Question 2

The product of two Gaussian random variables is distributed, in general, as a linear combination of two Chi-square random variables:

$XY \,=\, \frac{1}{4} (X+Y)^2 - \frac{1}{4}(X-Y)^2$

Now, $X+Y$ and $X-Y$ are Gaussian random variables, so that $(X+Y)^2$ and $(X-Y)^2$ are Chi-square distributed with 1 degree of freedom.

If $X$ and $Y$ are both zero-mean, then

$XY \sim c_1 Q - c_2 R$

where $c_1=\frac{Var(X+Y)}{4}$ , $c_2 = \frac{Var(X-Y)}{4}$ and $Q, R \sim \chi^2_1$ are central.

The variables $Q$ and $R$ are independent if and only if $Var(X) = Var(Y)$ .

In general, $Q$ and $R$ are noncentral and dependent.

Answer