The number of customers that arrive at a service station during a time t is a Poisson

random variable with parameter $\displaystyle \beta t$. The time required to service each customer is an exponential random variable with parameter a.

Identify the distribution of the number of customers N that arrive during the service time T of a specificc customer by:

finding the MGF of N

$\displaystyle f(t)=ae^{-at}$

$\displaystyle N(n)=\frac{\beta^n t^n}{n!}}$$\displaystyle e^{-\beta t}$

I have found the respective moment generating functions which are:

$\displaystyle M_f (s)=$$\displaystyle \frac{a}{a-s}$ for the exponential distribution and

$\displaystyle M_N (s)=$$\displaystyle e^{\beta t{e^s-\beta t}$

Now i have no idea how to combine the results to find the required PDF.

And i cant find anything in my notes about this.