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

random variable with parameter . 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

I have found the respective moment generating functions which are:

for the exponential distribution and

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.