MHB MGF relating to random sum of random variables

[nedflanders]

Hi all

I am doing this question right now and I don't even know how to start it up.
I know that it's in relation to a sum of a random number of random variables, but I don't know how to continue on from that.

I've read my textbook and it states some definition for an MGF which is:
$M_{y}(t) = \sum_{n=0}^{\infty} M_{X}(t)^n p_{N}(n)$ but it doesn't derive it, however, I think it relates to this question?

The question is as follows:

Thanks for the help

 

[Siron]

The random variable $N$ is defined as the number of customers arriving during the service time $T$ of a customer. If $t$ was a constant you could just apply the MGF of the Poisson distribution. But now $T$ is a random variable which has a Gamma distribution so we have to take that into account. The MGF of $N$ is defined as
$$\mbox{MGF}_{N}(k) = \mathbb{E}[e^{kN}] = \sum_{n=0}^{\infty} e^{kn} \mathbb{P}(N=n)$$

The only thing we have to do now is to calculate $\mathbb{P}(N=n)$ which is the probability that $n$ customers will arive at a service station during the service time $T$.

Any thoughts?