What is the pdf of $X/Y$, where $X$ has a symmetrical uniform distribution, and $Y$ is normal with mean zero?

Question

If X has a uniform distribution, between some -L and L, and Y has a normal distribution, with zero mean and variance=Sigma^2.

The distribution of Q=Y/X is called the "Slash distribution". https://en.wikipedia.org/wiki/Slash_distribution

How to obtain the distribution of Z=X/Y?

Thanks!

Answer 1

You already know $Y/X$ has pdf $f(q):=\frac{\varphi(0)-\varphi(q)}{q^2},\,\varphi(t):=\frac{1}{\sqrt{2\pi}}\exp-\frac{t^2}{2}$. Thus $Z:=\frac{1}{Q}$ has pdf $\frac{1}{z^2}f(\frac{1}{z})=\varphi(0)-\varphi(\frac{1}{z})=\frac{1}{\sqrt{2\pi}}\left(1-\exp-\frac{1}{2z^2}\right)$.