# MGF

• Jul 29th 2011, 09:36 AM
CountingPenguins
MGF
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 $\displaystyle \beta = \mu$.

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

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?
• Jul 29th 2011, 02:11 PM
mr fantastic
Re: MGF
Quote:

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 $\displaystyle \beta = \mu$.

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