We are given statistical sample of $X=(X_1,X_2,X_3)$, where $X_i\sim\Gamma(\alpha_i,1)$ and independent. Let $Z=X_1+X_2+X_3$ and $T$ three dimensional statistic $T:=(\frac{X_1}{Z},\frac{X_2}{Z},\frac{X_3}{Z})$.
We are asked to check are $T$ and $Z$ independent.
Hint: Find joint density function of $T$.
So far, I manage to find that:
$$Z\sim\Gamma(\alpha_1+\alpha_2+\alpha_3,1)$$ and that:
$$\frac{X_i}{Z}\sim\beta'(\alpha_i,\alpha_1+\alpha_2+\alpha_3)$$.
So:
$$f_T(y_1,y_2,y_3)=\prod_{i=1}^{3}f_{\frac{X_i}{Z}}(y_i).$$
However, now when I have to prove the independence of $T$ and $Z$ I get lost. How should I use this joint distribution function? Should I try to deduce something special about this distribution? Please, help.