120. The simple Poisson process of Section 3.6 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 [1,1 + Ar] is a(t) Ato(At). It can then be shown that the number of events occurring during an interval [1, 1] has a Poisson distribution with parameter = "(1) dt 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 a(t) = + 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. 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]?