MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

## Which iid variables give a normal variable, when multiplied?

Hello, I hope you'll find my riddle interesting.

Z = XY
Z ~ N(0, 1)
X, Y are iid random variables (independent, identically distributed). We assume X and Y are symmetric.
What is the distribution of X and Y?

It will be enough to find the answer for Z(+) = X(+) Y(+), where Z(+)'s distribution is the positive half of N(0,1), and X(+) and Y(+) are positive.

I am interested in knowing the pdf of X,Y, if it has a closed form.

-

## 1 Answer

First thought is to take U=ln[Z(+)]= ln[X(+)]+ln[Y(+)]. Work out the characteristic function of U and then square root it to get the characteristic function of ln[X(+)]. Sure the intergrals will be horribly messy though.

-
 Yes, I thought about doing it like that, but there is calculating the inverse transform of sqrt(Gamma(...)) on the way. – Arkadiusz Paterek Nov 27 2009 at 16:03