I am having trouble with this problem, I am trying to use the MGF but this may not be the best way to solve it. Any help would be great.
Suppose $X_1, ···, X_n \sim \mathcal{Exp}(\lambda)$ are independent. What is the distribution of $X_1/S$ where $S= X_1 + X_2 +···+ X_n$?
As @StubbornAtom's link noted, If $X\sim\Gamma(\alpha,\,1),\,Y\sim\Gamma(\beta,\,1)$ are independent then $Z:=\frac{X}{X+Y}\sim\operatorname{B}(\alpha,\,\beta)$. Take $X:=\frac{X_1}{\lambda},\,Y:=\frac{S-X_1}{\lambda}$ so $\alpha=1,\,\beta=n-1$. So if $n\ge2$, the PDF of $Z$ is $(n-1)(1-z)^{n-2}$ on its support $[0,\,1]$. This has mean $\frac{\alpha}{\alpha+\beta}=\frac1n$, which is obvious because, although not independent, the $\frac{X_i}{S}$ each have the same mean, $\frac1n$ times that of their sum, which is a constant variable equal to $1$.
