I have two i.i.d. random variables $X$ and $Y$ and I want to generate a set of $n$ random samples from the distribution of the ratio between the two variables $Z=X/Y$.
The pdf of $Z$ is described by: $$ f_Z(t) = \int_{-\infty}^\infty |u| ~ f_{X,Y}(tu,u)~\mathrm{d} u, $$ and then its cdf: $$ F_Z(t) = \int_{-\infty}^t f_Z(u)~\mathrm{d} u. $$ Then, in theory I can generate random samples $z\sim Z$ by: $$ z=F^{-1}_Z(r), $$ where $r$ is a uniform random sample in $[0,1]$.
The problem is that obtaining $f_Z(t)$ is not straightforward, specially when $X$ and $Y$ are both Weibull distributions; neither it is to obtain $F_Z(t)$. Is there any way to obtain random samples from $Z$ by directly using random samples from the marginal distributions of $X$ and $Y$ ?