Finding the pdf of $X_1/(X_1+X_2)$ given $X_1,X_2 \sim \operatorname{Exp}(1)$ I have that $X_1,X_2 \sim \operatorname{Exp}(1)$.
I need to find the pdf (probability density function) of $T$ where $T= X_1 + X_2$ and $R= X_1/(X_1+X_2)$.
I convolved the pdf's of $X_1$ and $X_2$ to get $t\exp(-t)$ for the pdf of $T$, but I'm completely lost on how to start with R's distribution.
Is there a way I could get a joint pdf of $T$ and $R$, then integrate $T$ out to get $R$'s marginal?
I can't really think of a way to solve for it directly.

Hint:
For $r\in(0,1)$
$$P\left(R<r\right)=\int_{0}^{\infty}P\left(R<r\mid X_{1}=x\right)f_{X_{1}}\left(x\right)dx$$
