Math

StatisticsProbability and Statistics for Engineering and the SciencesThe 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 + Δ t ] is α ( t ) · Δ t + o (Δ t ). It can then be shown that the number of events occurring during an interval ( t 1 , t 2 ] has a Poisson distribution with parameter μ = ∫ t 2 t 1 α ( t ) d t 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 + b 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]?BuyFind*arrow_forward*

9th Edition

Jay L. Devore

Publisher: Cengage Learning

ISBN: 9781305251809

Chapter 3, Problem 120SE

Textbook Problem

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* + Δ*t*] is *α*(*t*) · Δ*t* + *o*(Δ*t*). It can then be shown that the number of events occurring during an interval (*t*_{1}, *t*_{2}] 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

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]?

Probability and Statistics for Engineering and the Sciences

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