120. 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



t2 t1

(t) 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

(t)
eabt 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]?