$X$ and $Y$ are independent standard uniform random variables. What is the density of $Z = X/Y$?
So far I have:
$f_X(x) = f_Y(y) = 1\text{ if }0 \le x,y \le 1$
$\begin{align} f_Z(z) & = \int_{-\infty}^\infty f_X(zx)f_Y(x)|x| dx \\ & = \int_{-\infty}^\infty f_X(zx)f_Y(x)x dx \\ & = \int_{-\infty}^\infty f_X(zx)x dx \\ \end{align}$
I think that $f_X(zx)$ in the integrand can be replaced with $1$ but how would I change the bounds of the integral?