Ratio of Gamma random variables If $X_i$, $i=1,2$ are independent gamma$(\alpha_i,1)$ random variables, find the distribution of $\frac{X_1}{X_1+X_2}$ and $\frac{X_2}{X_1+X_2}$.
Attempt:
Let $Y_1 = \frac{X_1}{X_1+X_2}$ and $Y_2=X_1+X_2$.
Then $X_1 = Y_1Y_2$ and $X_2 = Y_2-Y_1Y_2$.
The Jacobian matrix is :
$\hspace{15mm}J = \left| \begin{array}{ccc}
y_2 & y_1 \\
-y_2& 1-y_1 \\
 \end{array} \right| = y_2$
I'm having trouble plugging the values back in to the gamma equation. Since we want to know the distribution of $Y_1$, do we first find the joint distribution of $f_{y_1,y_2}(y_1,y_2)$ and then integrate $y_2$ out?

Here How exactly are the beta and gamma distributions related? you can find in one of the solutions posted. X1/X1+X2 has betta distribution
