Distribution of ratio of sums of gamma distributions

If $X_1,X_2$ are independent r.v.s with $X_1 \sim \Gamma(\alpha,\theta)$, $X_2 \sim \Gamma(\beta,\theta)$ then it is known that $$\frac{X_1}{X_1+X_2} \sim \text{Beta}(\alpha,\beta)$$ Let $X_i$ be iid with $X_i \sim \Gamma(\alpha,\theta)$. What is the distribution of $$\frac{X_1}{\sum_{i=1}^n X_i}$$ ? What about the special case where $\alpha=2$, $\theta=1$?

EDIT: Matched question to did's answer :)

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Hint: $X_2+\cdots+X_n$ is $\Gamma((n-1)\alpha,\theta)$. (Hence the special case is Beta$(2,2n-2)$.) – Did Sep 24 '12 at 23:03
Didier, that's an answer, not a hint! – Dilip Sarwate Sep 25 '12 at 2:02
Thanks, did! Ouch, that was obvious. – Haderlump Sep 25 '12 at 9:00
@DilipSarwate I obeyed your command. :-) – Did Sep 25 '12 at 9:12

1 Answer

This follows from the $n=2$ case since the distribution of $X_2+\cdots+X_n$ is $\Gamma((n-1)\alpha,\theta)$. (Hence the special case is Beta$(2,2n-2)$.) Note that none of this depends on $\theta$, as long as this parameter stays the same for every $X_k$.

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