Say $X_1,X_2$ are independently drawn from the same distribution (call it $X$) and that their product, $X_1X_2$ falls on a standard normal distribution.
Is it possible to get a pdf or cdf for $X$?
My progress: The $n$th moment of a standard normal is $0$ for odd $n$ and $n!!$ for even $n$. Then for even $n$:
$\mathbb{E}[(X_1 X_2)^n] = \mathbb{E}[X_1^n] \mathbb{E}[X_2^n] = \mathbb{E}[X^n]^2 = n!! $
Thus the $n$th moment of $X$ is $\mathbb{E}[X^n] = \sqrt{n!!}$ for even $n$ and zero otherwise. Therefore...