Even though this question has been answered somewhat here, here, here, I'd like to ask what the probability density function of $z = x y$ is, where $x \sim \mathcal{U}[-1,1]$, $y \sim \mathcal{N}(0,1)$, and $z$ are all univariate. Furthermore, what are the mean and variance of $z$? (By symmetry, it is easy to see $\mathbb{E}[z] = 0$, but I want a more rigorous proof.)
Edit: What if $x \sim \mathcal{U}[-\theta,\theta]$?
Since $|X|,\,|Y|$ have respective PDFs $1_{[0,\,1]}(x),\,\sqrt{\tfrac{2}{\pi}}e^{-y^2/2}1_{[0,\,\infty)}(y)$, $|Z|=|X||Y|$ has on-$[0,\,\infty)$ PDF $\int_0^1x^{-1}\sqrt{\tfrac{2}{\pi}}e^{-z^2/(2x^2)}dx=\tfrac{1}{\sqrt{2\pi}}\Gamma\left(0,\,\tfrac12z^2\right)$ in terms of the upper incomplete Gamma function. Since $X,\,Y$ have symmetric distributions, $Z$'s PDF on $\Bbb R$ is $\tfrac{1}{\sqrt{8\pi}}\Gamma\left(0,\,\tfrac12z^2\right)$.