Exercise 4.51. Suppose customer arrivals at a post office are modeled by a Poisson process N with intensity λ > 0. Let T1 be the time of the first arrival. Let t > 0. Suppose we learn that by time t there has been precisely one arrival, in other words, that Nt = 1. What is the distribution of T1 under this new information? In other words, find the conditional probability P(T1 ≤ s|Nt = 1) for all s ≥ 0. Hint. You should see a familiar distribution.