It is given that $X_i \sim^{\text{ind}} \text{Gamma}(\alpha,p_i)$ Find the distributions of $Y_i=\frac{X_i}{X_1+X_2+...+X_i}$, where $i=2,3,..k$

Question

It is given that $X_i \sim^{\text{independent}} \text{Gamma}(\alpha,p_i)$

Find the distributions of $Y_i=\frac{X_i}{X_1+X_2+...+X_i}$, where $i=2,3,..k$

I have done it for $k=3$, by doing the transformation $X_1=Y_1, X_2=\frac{Y_1 Y_2}{1-Y_2}, X_3=\frac{Y_1Y_3}{(1-Y_2)(1-Y_3)}$ But the Jacobian calculation and joint pdf is quite tedious to find. It will be more difficult to find for $k$ by this procedure. Please help.