0
$\begingroup$

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?

$\endgroup$
2
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.