Statistical model with $\Gamma(\alpha_i,1)$ sample

Question

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.

By the way, I do see that $T_1+T_2+T_3=1$. Is there a way to use that?

Answer 1

This post has self study tag so I will just describe the way to solve this problem.

The joint distribution of $T$ is actually the joint distribution of $Y_1 = \frac{X_1}{Z}$, $Y_2 = \frac{X_2}{Z}$ because the third coordinate is automatically set to be $1-Y_1-Y_2$.

To get the distribution of $T$, I guess you should firstly use transformation like : $Y_1 = \frac{X_1}{Z}$, $Y_2 = \frac{X_2}{Z}$, $Y_3 = Z$ and secondly get the joint distribution of these variables and finally integrate it with $y_3$ to get the marginal distribution of $Y_1$ and $Y_2$ which is the distribution of $T$.

This can be done with a little effort. You can use the fact that $X_1,X_2,X_3$ are independent of each others and the determinant of Yacobian matrix isn't complex.

And actually you don't have to integrate with $y_3$ to show the independence. Before integration, you can figure out that the joint pdf $g_{Y_1,Y_2,Y_3}$ can be expressed as the multiplication form like $g_{Y_1,Y_2}(y_1,y_2) g_{Y_3}(y_3)$ which means that $Z=Y_3$ is independent of $T$.

I am not a native English speaker so please don't mind my awkward expressions and it would be appreciated if you improve my sentences. Thank you.

This is the way to find the Drichlet distribution which is mentioned by the comments of @GordonSmyth.