Derive density of $Z=XY$, $X\sim U(0,1)$ and $Y\sim\mathcal N(0,1)$. I have been stumped for a few days on this. I have two random variables $X\sim U(0,1)$ and $Y\sim\mathcal N(0,1)$, which are independent. How can I get the density of $Z = XY$?
I put $Z = XY, W = Y$ i.e. $X=Z/W, Y=W$ and achieved the Jacobian as $J={1\over|W|}$,
but I've got
$f_{Z,W}(z,w)={1\over\sqrt{2\pi}|w|}{\exp(-w^2/2)}$
and I don't know how to integrate this w.r.t $w$ and get the (marginal) density of $Z$.
Could you please help me with this problem? I tried partial integration too but it doesn't work.

angryavian's comment seems to be the way to get a close to numerical answer.
Incase you were interested visually in the density here is a plot with $10^7$ samples. It seems to be strangely convex, my guess is that it would have looked very normal.

