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 :)