Say I have X∼N(a,b) and Y∼N(c,d). Is XY also normally distributed?
Is the answer any different if we know that X and Y are independent?
The product of two Gaussian random variables is distributed, in general, as a linear combination of two Chi-square random variables:
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
where c1=Var(X+Y)4, c2=Var(X−Y)4 and Q,R∼χ21 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.