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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 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]?

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Probability and Statistics for Eng...

9th Edition
Jay L. Devore
Publisher: Cengage Learning
ISBN: 9781305251809
BuyFind

Probability and Statistics for Eng...

9th Edition
Jay L. Devore
Publisher: Cengage Learning
ISBN: 9781305251809

Solutions

Chapter
Section
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 + Δt] is α(t) · Δt + ot). It can then be shown that the number of events occurring during an interval (t1, t2] 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.

  1. a. What is the expected number of events in the interval [0, 4]? In [2, 6]?
  2. b. What is the probability that at most 15 events occur in the interval [0, .9907]?

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Chapter 3 Solutions

Probability and Statistics for Engineering and the Sciences
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Ch. 3.2 - Airlines sometimes overbook flights. 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In a Poisson process, what has to happen in...Ch. 3 - Consider a deck consisting of seven cards, marked...Ch. 3 - After shuffling a deck of 52 cards, a dealer deals...Ch. 3 - The negative binomial rv X was defined as the...Ch. 3 - Of all customers purchasing automatic garage-door...Ch. 3 - In some applications the distribution of a...Ch. 3 - A k-out-of-n system is one that will function if...Ch. 3 - A manufacturer of integrated circuit chips wishes...Ch. 3 - Of the people passing through an airport metal...Ch. 3 - An educational consulting firm is trying to decide...Ch. 3 - Consider a disease whose presence can be...Ch. 3 - Let p1 denote the probability that any particular...Ch. 3 - The purchaser of a power-generating unit requires...Ch. 3 - A plan for an executive travelers club has been...Ch. 3 - Forty percent of seeds from maize (modern-day com)...Ch. 3 - A trial has just resulted in a hung jury because...Ch. 3 - A reservation service employs five information...Ch. 3 - Grasshoppers arc distributed at random in a large...Ch. 3 - A newsstand has ordered five copies of a certain...Ch. 3 - Individuals A and B begin to play a sequence of...Ch. 3 - A test for the presence of a certain disease has...Ch. 3 - The generalized negative binomial pmf is given by...Ch. 3 - There are two Certified Public Accountants in a...Ch. 3 - The mode of a discrete random variable X with pmf...Ch. 3 - A computer disk storage device has ten concentric...Ch. 3 - If X is a hypergeometric rv, show directly from...Ch. 3 - Use the fact that allx(x-)2p(x)x:x-k(x-)2p(x) to...Ch. 3 - The simple Poisson process of Section 3.6 is...Ch. 3 - Consider a collection Al,... , Ak of mutually...Ch. 3 - Consider a communication source that transmits...

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