Here’s an interesting result: Let , i.e. and assume that is a continuous random variable such that follows distribution. What is the distribution of ?
For we have
so that .
Note that if is a Poisson process with intensity and is a Bernoulli process with parameter , then we can consider the split processes , , where are the arrival times of the original process, and is the number of arrivals in with , the number of arrivals in with . It is well-known that and are independent Poisson processes with intensity and , respectively. So is the interarrival distribution of .