136. The simple Poisson process of Section 3.7 is characterized by a constant rate a 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 a(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. Statist. Assoc., 1989: 360—372, considers the intensity function as appropriate for events involving transmission of HIV (the AIDS virus) via blood transfusions.

Suppose that a2 and b.6 (close to values suggested in the paper), with time in years.

a. What 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]?