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 $i=2,3,..k$

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.

• You should use the transformation $Y_1=X_1+X_2+X_3\,,Y_2=\frac{X_2}{X_1+X_2}$ and $Y_3=\frac{X_3}{X_1+X_2+X_3}$ for $k=3$. – StubbornAtom Feb 17 '18 at 5:52
• I had posted this question here on CV. – StubbornAtom Feb 17 '18 at 5:55
• Ok, so taking $Y_1= \sum_{i=1}^{k}X_i$ seals the deal – Legend Killer Feb 17 '18 at 5:58
• Your $X_i$'s can only be independent and not i.i.d. as you have different parameters $p_i$'s, unless of course the $p_i$'s are equal. – StubbornAtom Feb 17 '18 at 6:00