Let $X$ follow a uniform distribution and $Y$ follow a normal distribution. What can be said about $\frac X Y$? Is there a distribution for it?
I found the ratio of two normals with mean zero is Cauchy.
Let random variable $X \sim Uniform(a,b)$ with pdf $f(x)$:
where I have assumed $0<a<b$ (this nests the standard $Uniform(0,1)$ case). [ Different results will be obtained if say parameter $a<0$, but the procedure is exactly the same. ]
Further, let $Y \sim N(\mu, \sigma^2)$, and let $W=1/Y$ with pdf $g(w)$:
Then, we seek the pdf of the product $V = X*W$, say $h(v)$, which is given by:
where I am using mathStatica's
Here are two plots of the pdf:
Monte Carlo check
Here is a quick Monte Carlo check of the Plot 2 case, just to make sure no errors have crept in:
The blue line is the empirical Monte Carlo pdf, and the red dashed line is the theoretical pdf $h(v)$ above. Looks fine :)
Besides the reciprocal of the slash distribution (or @Glen_b's "backslash distribution!"), a kind of ratio distribution, I don't know what to call it either, but I'll simulate one version in R.