The simple Poisson process of Section 3.6 is characterized by a constant rate ? at which events occur per unit time. A generalization of this is to suppose that the probability of exactly one event occurring in the interval[ t,t + ?t] is ?(t). ?t + o(?t). It can then be shown that the number of events occurring during an interval [t1, t2] has a Poisson distribution with parameter The occurrence of events over time in this situation is called a nonhomogeneous Poisson process. The article “Inference Based on Retrospective Ascertainment,” J. Amer. Stat. Assoc., 1989: 360–372, considers the intensity function ?(t) = e a+bt as appropriate for events involving transmission of HIV (the AIDS virus) via blood transfusions. Suppose that a = 2 and b = .6(close to values suggested in the paper), with time in years. a.? hat is the expected number of events in the interval [0, 4]? In [2, 6]? b.? ?What is the probability that at most 15 events occur in the interval [0, .9907]?

Answer Step 1 of 4 Given (t) = e a+bt, a=2, b=0.6 2+0.6t Then (t)=e Step 2 of 4 a)The expected number of events in the interval [0, 4] t2 = (t) dt t1 4 = 2+0.6dt 0 0.6t2 4 =[e2 + 0.6t(2t + 2 ]0 =123.44 The expected number is 123