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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?

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    $\begingroup$ Do you need to add a self-study tag to this question? $\endgroup$ – Gordon Smyth Jun 26 '17 at 7:27
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    $\begingroup$ Hint: your first two statements are correct (distributions for $Z$ and $X_i/Z$) but your third statement is wrong (distribution of $T$). Obviously the $X_i/Z$ add up to 1 so they can't be independent. Have you studied Dirichlet distributions? $\endgroup$ – Gordon Smyth Jun 26 '17 at 7:33
  • $\begingroup$ So, we can say that because $T_1+T_2+T_3=1$ there is no PDF? $\endgroup$ – Aidas Jun 26 '17 at 7:38
  • $\begingroup$ Certainly not! Let me ask again: have you studied Dirichlet distributions? $\endgroup$ – Gordon Smyth Jun 28 '17 at 0:06
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    $\begingroup$ This is a hard self-study question if you don't know the Dirichlet distribution. Also, I don't believe that $Z$ and $T$ are independent, unless all the $\alpha_i$ are equal. $\endgroup$ – Gordon Smyth Jun 28 '17 at 7:42
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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.

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