If $r = x/y$ is the ratio of two independent continuous positive random variables, its distribution can be estimated by generating random samples from the distribution of $x$ and $y$, and then proceeding in various ways. It is shown, using well-known results in the theory of Wilcoxon's test that the uniformly minimum variance unbiased estimate of $H(A) = P(r \leqq A)$ is obtained by computing Wilcoxon's statistic for the random variables $u_i = x_i, v_i = Ay_i(i = 1, \cdots, N)$. The variance of the estimate of $H(A)$ is readily estimated. The computations required by this approach are more arduous than those needed to estimate $H(A)$ from the quantities $r_i = x_i/y_i$, but may be worthwhile where the major part of the computations lies in generating the $x_i$ and $y_i$. Extension of the reasoning leads to choosing different numbers of $x$'s and $y$'s if they are of different complexity to generate. Further, if the distribution of one of the quantities $x$ or $y$ is known then an effectivity infinite sample from that population is already available and the distribution of $r$ can be estimated by sampling only the variable with unknown distribution, which may (or may not) result in economy of effort.
"Use of Wilcoxon Test Theory in Estimating the Distribution of a Ratio by Monte Carlo Methods." Ann. Math. Statist. 33 (3) 1194 - 1197, September, 1962. https://doi.org/10.1214/aoms/1177704483