## Poisson distribution

The number of customers that arrive at a service station during a time t is a Poisson
random variable with parameter $\beta t$ . 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 speci ficc customer by:

finding the MGF of N

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

I have found the respective moment generating functions which are:

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

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

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