laws of joint distribution of several random variables

Question

How to solve problems of this type? random variables X and Yare independent and have a density $\mathbb{I}_{[0,\infty]}g(x)$. Obtain an explicit formula and plot the density of a random variable $Z=Y/(X+Y)$

Answer 1

Hint

One approach that sometimes works is to find $$ \begin{split} F_Z(z) &= \mathbb{P}[Z \le z] \\ &= \mathbb{P}\left[\frac{Y}{X+Y} \le z\right] \\ &= \mathbb{P}\left[Y \le zX + zY\right] \\ &= \mathbb{P}\left[zX + (z-1)Y \ge 0\right] \\ &= \iint_D f(x,y) \ dxdy, \end{split} $$ where $D$ is the subset of the plane such that $zx + (z-1)y \ge 0$.

You know $f(x,y)$, can you finish this?