How to find the first order PDF of is left as an exercise. Note that is the sum of independent and identically distributed random variables.

Example 2

The number of customers arriving at a service station is a Poisson process with a rate 10 customers per minute.
(i) What is the mean arrival time of the customers?
(ii) What is the probability that the second customer will arrive 5 minutes after the first customer has arrived?
 (iii) What is the average waiting time before the 10 th customer arrives?