[Math] Find the moment generating function of the sum of exponential random variables $S=X_1+X_2+X_3+X_4$

Question

Let $X_1+X_2+X_3+X_4$ be iid exponential random variables with parameter λ, and $S=X_1+X_2+X_3+X_4$
S follows the gamma distribution with parameters $\lambda$ and $r=4$.
We know that an exponential random variable X with parameter $\lambda$ has moment
generating function
$E[e^{tX}]=\frac{\lambda}{\lambda-t}$ if $t<\lambda$ and $+\infty$ if $t \geq \lambda$
a) find MGF $M_S(t)$ (don't forget to declare the domain)
b) find the 1st and 2nd moment of S, then find the variance
I know that for b, you can just find $M_S'(0)$ and $M_S''(0)$ but for a, I'm not sure on how to get started on finding the moment generating function

Answer 1

The mgf of a sum of independent random variables is the product of the individual mgf.

In our case each $X_i$ has mgf $\frac{\lambda}{\lambda-t}$, and therefore the sum has mgf $\left(\frac{\lambda}{\lambda-t}\right)^4$ for $t\lt \lambda$.

You know how to do b) using the mgf. As a check, note that the mean of $S$ is the sum of the means of the $X_i$, and the variance of $S$ is the sum of the variances of the $X_i$.