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

Say I have XN(a,b) and YN(c,d). Is XY also normally distributed?

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

Answer

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

XY=14(X+Y)214(XY)2

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

If X and Y are both zero-mean, then

XYc1Qc2R

where c1=Var(X+Y)4, c2=Var(XY)4 and Q,Rχ21 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.

Attribution
Source : Link , Question Author : jamaicanworm , Answer Author : Ulisses Braga-Neto

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