If $X_i \sim \text{Exp}(\theta)$ what is the distribution of $\frac{n}{X_1 + \dots + X_n}$?

Question

If $X_1, \dots X_n$ are independent and Exponentially distributed with parameter $\theta$, then by examining the Characteristic function (or Moment generating function) of the sum $$ X_1 + \dots + X_n $$ one sees that the sum follows a Gamma distribution with parameters $n$ and $\theta$.

I wonder if similarly one may derive the distribution of $$ \frac{n}{X_1 + \dots + X_n} \quad ? $$

Most grateful for any help provided!

Answer 1

Very easy:

if (as it is) $Y=\Sigma_i X_i\sim Gamma(n;\theta)$ then $\frac{1}{Y}\sim \text{Inverse Gamma}$

thus the law of $\frac{n}{Y}$ can be derived immediately by a simple transformation