Let $X$, $Y$, $Z$, and $W$ be i.i.d. copies of a standard gaussian variable, that is in distribution $\mathcal{N}\left(0,1\right)$, then what is the distribution of $\left\frac{XY}{Z}W\right$?
Thanks!
Let $X$, $Y$, $Z$, and $W$ be i.i.d. copies of a standard gaussian variable, that is in distribution $\mathcal{N}\left(0,1\right)$, then what is the distribution of $\left\frac{XY}{Z}W\right$? Thanks! 


I don't think there's any reason for it to have a nice distribution. After fooling around with it a bit, the following seem to be true: *It has infinite variance (Not too hard to show, X/Y has infinite variance, and the other random variables just increase it) *Pretty sure it has an infinite mean (Empirically, the samplemean seems to be stationary, and theoretically, there's a bounding argument with a Cauchy that I'm not positive of. Any specific questions about the distribution? 


Yes, the mean is infinite. (If it was finite, since $W$ is integrable, $XY/Z$ would be integrable. Since $XY$ and $Z$ are independent and $XY$ is not identically $0$, $1/Z$ would be integrable. But now $Z$ is Gaussian, its density around $0$ is equivalent to a positive constant, hence $1/Z$ is not integrable.) Why is this random variable, or its distribution, interesting? 


Yes it can: the probability that $(XY/Z)W\ge t$ is bounded by $c/t+o(1/t)$ with $c^2=2/\pi^3$, when $t\to\infty$. (By the way, Yemon Choi DID NOT suggest that the tail estimate might be interesting, he asked why you asked. And got no answer.) 

