Originally Posted by

**CountingPenguins** The question is:

Let $\displaystyle X_{1}, X_{2}, ... X_{n}$ be independent and identically distributed exponential random variables each with mean $\displaystyle \mu$. Find the MGF of $\displaystyle \Sigma_{i=1}^n X_{i}$ and verify that it is the MGF of a gamma random variable with parameters $\displaystyle \alpha$ = n and $\displaystyle \beta = \mu$.

The answer is:

$\displaystyle M_{\Sigma X_{i}}(\Theta) = \dfrac{1}{(1-\dfrac{\Theta}{\gamma})^{n}}, \Theta < \gamma$

I have the MGF for an exponential distribution and the MGF for a gamma distribution and the answer, but I cannot see how to put it all together. Any ideas?