The simple Poisson process of Section 3.6 is characterized by a constant rate α 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 + Δt] is α(t) ? Δt + o(Δt). 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
α(t) = eα + 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. 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]?

(t) dt