Estimating PDF of ratio of random variables

Question

I have a step in my project in which I have two random variables $U$ and $V$ with their distributions, and I need to calculate the quantity $\frac{U}{U+V}$. Unfortunately, this ratio does not have an analitically tractable distribution.

My question is: is it possible to compute the PDF of the ratio by taking samples from the numerator, taking samples from the denominator, computing the ratio and then using the histogram of the ratios as the estimate for the PDF?

Thank you very much!

Answer 1

Graphical comment: Using R.

u = runif(10^6); v = runif(10^6)
x = u/(u+v)
hist(x, prob=T, br=50)

You can derive the PDF of $X = \frac{u}{U+V}$ by using a bivariate transformation.