The simple Poisson process of Section 3.6 is characterized by a constant rate α at which...

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 + ?t] is α(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

α(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. 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]?