Question

Suppose that telephone calls arriving at a particular switchboard follow a Poisson process with an average of 5 calls coming per minute. What is the probability that up to a minute will elapse by the time 2 calls have come in to the switchboard?

Answer 1

Solution

The Poisson process applies, with time until 2 Poisson events following a gamma distribution with β = 1/5 and α = 2. Denote by X the time in minutes that transpires before 2 calls come. The required probability is given by

While the origin of the gamma distribution deals in time (or space) until the occurrence of α Poisson events, there are many instances where a gamma distri-bution works very well even though there is no clear Poisson structure. This is particularly true for survival time problems in both engineering and biomedical applications.