Let $X \sim \Gamma(k,\theta)$ and Y $\sim \Gamma(p,\theta)$
What is the distribution of $Z = \dfrac{X}{X+Y}$
I know that there are some similar questions here on Stack, but these questions are related to the case when the Gammas are independent and the result is a Beta Prime Distribution, but in this case I cannot simply sum the two Gammas on the denominator and use this kind of result.
I tried to do the Jacobian method, but I struggled to find the joint distribution of X and Y and ended up with a big expression that didn't seems to be correct.
Let $z\in(0,1)$. Then, \begin{align*} F_Z(z) = \mathbb P(Z\le z) = \mathbb P(X \le z(X+Y)) = \mathbb P\!\left(\frac{X}{Y} \le \frac{z}{1-z}\right). \end{align*} Now use the fact that $X/Y$ follows a Beta Prime Distribution.