Suppose I have $X_i \sim \operatorname{Exp}\left(\beta\right)$. Then, the sample mean of these random variables has gamma distribution (source).

$$\overline{X} \sim \operatorname{Gamma}\left( n, \dfrac{\beta}{n} \right)$$

Why is this the case? I understand that the exponential distribution is a special case of the gamma distribution. How can I prove that the sample mean is distributed as such?


The sum of $n$ IID exponential random variables with rate $\beta$ is well known to follow a gamma distribution with parameters $n, \beta$. See: Gamma Distribution out of sum of exponential random variables.

I assume you can prove the following:
Let $c>0$ and let $X\sim \operatorname{Gamma}(r,\lambda)$ with $r,\lambda>0$. It is well-known that $cX\sim\operatorname{Gamma}(r, \lambda/c)$.

Using these two, you can prove your desired result.


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