Let $X \simeq \Gamma(a,1)$ and $Y \simeq \Gamma(b,1)$ be independent. Show that $U = X +Y$ and $V = X/(X +Y)$ are independent and find their distributions.
How would you do this for $U$ and $V$, do you use the Jacobian matrix transform to prove this? Additionally, when you're finding their distributions are you looking for the joint distributions of $U$ and $V$ or the distribution of $U$ and the distribution of $V$?