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Chapter 3: Q120E (page 139)

The simple Poisson process of Section \({\rm{3}}{\rm{.6}}\) is characterized by a constant rate a 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 \({\rm{(t,t + \Delta t)}}\) is \({\rm{\alpha (t)}} \cdot {\rm{\Delta t + o(\Delta t)}}\). It can then be shown that the number of events occurring during an interval \({\rm{(}}{{\rm{t}}_{\rm{1}}}{\rm{,}}{{\rm{t}}_{\rm{2}}}{\rm{)}}\) has a Poisson distribution with parameter

\({\rm{\mu = }}\int_{{{\rm{t}}_{\rm{2}}}}^{{{\rm{t}}_{\rm{1}}}} {{\rm{\alpha (t)dt}}} \)

The occurrence of events over time in this situation is called a nonhomogeneous Poisson process. The article 鈥淚nference Based on Retrospective Ascertainment,鈥� (J. Amer. Stat. Assoc., \({\rm{1989: 360 - 372}}\)), considers the intensity function

\({\rm{\alpha (t) = }}{{\rm{e}}^{{\rm{a + bt}}}}\)

as appropriate for events involving transmission of HIV (the AIDS virus) via blood transfusions. Suppose that \({\rm{a = 2}}\) and \({\rm{b = }}{\rm{.6}}\) (close to values suggested in the paper), with time in years.

a. What is the expected number of events in the interval \({\rm{(0,4)}}\)? In \({\rm{(2,6)}}\)?

b. What is the probability that at most \({\rm{15}}\) events occur in the interval \({\rm{(0,}}{\rm{.9907)}}\)?

Short Answer

Expert verified

(a) The expected number of events in the interval\({\rm{(0,4)}}\)is\({\rm{123}}{\rm{.436}}\)and in\({\rm{(2,6)}}\)is\({\rm{409}}{\rm{.82}}\).

(b) The probability that at most \({\rm{15}}\) events occur in the interval \({\rm{(0,}}{\rm{.9907)}}\) is \({\rm{0}}{\rm{.951}}\).

Step by step solution

01

Concept Introduction

A Poisson distribution is a probability distribution used in statistics to show how many times an event is expected to happen over a certain amount of time. To put it another way, it's a count distribution.

02

Number of Events

Proposition: For a random variable \({\rm{X}}\) with Poisson Distribution with parameter \({\rm{\mu > 0}}\), the following is true 鈥�

\({\rm{E(X) = V(X) = \mu }}\)

The same stands for the given nonhomogeneous Poisson Process. When we find the parameter\({\rm{\mu }}\)for the given interval we will find the expected values for each interval, where for an interval\({\rm{(}}{{\rm{t}}_1},{t_2})\)the parameter is 鈥�

\({\rm{\mu = }}\int_{{{\rm{t}}_{\rm{1}}}}^{{{\rm{t}}_{\rm{2}}}} {\rm{\alpha }} {\rm{(t)dt = }}\int_{{{\rm{t}}_{\rm{1}}}}^{{{\rm{t}}_{\rm{2}}}} {{{\rm{e}}^{{\rm{a + bt}}}}} {\rm{dt}}\)

The following holds 鈥�

\(\begin{aligned}\int_{{{\rm{t}}_{\rm{1}}}}^{{{\rm{t}}_{\rm{2}}}} {{{\rm{e}}^{{\rm{a + bt}}}}} dt &= \left| {\begin{array}{*{20}{c}}{{\rm{a + bt = x}}}&{{\rm{t = }}{{\rm{t}}_{\rm{1}}} \Rightarrow {\rm{x = a + b}}{{\rm{t}}_{\rm{1}}}}\\{{\rm{bdt = dx}}}&{{\rm{t = }}{{\rm{t}}_{\rm{2}}} \Rightarrow {\rm{x = a + b}}{{\rm{t}}_{\rm{2}}}}\end{array}} \right|\\ & = \int_{{\rm{a + b}}{{\rm{t}}_{\rm{1}}}}^{{\rm{a + b}}{{\rm{t}}_{\rm{2}}}} {\frac{{\rm{1}}}{{\rm{b}}}} {{\rm{e}}^{\rm{x}}}{\rm{dx}}\\&= \left. {\frac{{\rm{1}}}{{\rm{b}}}{{\rm{e}}^{\rm{x}}}} \right|_{{\rm{a + b}}{{\rm{t}}_{\rm{1}}}}^{{\rm{a + b}}{{\rm{t}}_{\rm{2}}}}\\ &= \frac{{\rm{1}}}{{\rm{b}}}\left( {{{\rm{e}}^{{\rm{a + b}}{{\rm{t}}_{\rm{2}}}}}{\rm{ - }}{{\rm{e}}^{{\rm{a + b}}{{\rm{t}}_{\rm{1}}}}}} \right)\end{aligned}\)

(a)

For interval\({\rm{(0,4),}}{{\rm{t}}_{\rm{1}}}{\rm{ = 0,}}{{\rm{t}}_{\rm{2}}}{\rm{ = 4}}\), and the given parameters\({\rm{a}}\)and\({\rm{b}}\)the following holds 鈥�

\(\begin{array}{c}{{\rm{\mu }}_{\rm{1}}}{\rm{ = }}\int_{\rm{0}}^{\rm{4}} {{{\rm{e}}^{{\rm{2 + 0}}{\rm{.6t}}}}} {\rm{dt = }}\frac{{\rm{1}}}{{{\rm{0}}{\rm{.6}}}} \cdot \left( {{{\rm{e}}^{{\rm{2 + 0}}{\rm{.6}} \cdot {\rm{4}}}}{\rm{ - }}{{\rm{e}}^{{\rm{2 + 0}}{\rm{.6}} \cdot {\rm{0}}}}} \right)\\{\rm{ = 123}}{\rm{.436}}\end{array}\)

Similarly, for interval\({\rm{(2,6)}}\), it is obtained 鈥�

\(\begin{array}{c}{{\rm{\mu }}_{\rm{2}}}{\rm{ = }}\int_{\rm{2}}^{\rm{6}} {{{\rm{e}}^{{\rm{2 + 0}}{\rm{.6t}}}}} {\rm{dt}}\\{\rm{ = 409}}{\rm{.82}}\end{array}\)

Therefore, the values obtained are \({\rm{123}}{\rm{.436}}\) and \({\rm{409}}{\rm{.82}}\).

03

Probability that at most fifteen events occur

(b)

Assume that we model the interval with random variable \({\rm{X}}\). In this case, we need to find parameter p for the random variable \({\rm{X}}\) and find probability of event \({\rm{X}} \le {\rm{15}}\) or equally \({\rm{F(15;\mu )}}\). The parameter \({\rm{\mu }}\) can be found as 鈥�

\(\begin{array}{c}{\rm{\mu = }}\int_{\rm{0}}^{{\rm{0}}{\rm{.9907}}} {{{\rm{e}}^{{\rm{2 + 0}}{\rm{.65}}}}{\rm{dt}}} \\{\rm{ = 9}}{\rm{.9996}}\end{array}\)

Approximate \({\rm{\mu }}\) with \({\rm{10}}\) and the requested probability is 鈥�

\({\rm{F(15;10) = 0}}{\rm{.951}}\)

Therefore, the value is obtained as \({\rm{0}}{\rm{.951}}\).

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Most popular questions from this chapter

Of all customers purchasing automatic garage-door openers, 75% purchase a chain-driven model. Let \({\bf{X}}{\rm{ }}{\bf{5}}\) the number among the next \({\bf{15}}\) purchasers who select the chain-driven model.

a. What is the pmf of \({\bf{X}}\)?

b. Compute \({\bf{P}}\left( {{\bf{X}}{\rm{ }}.{\rm{ }}{\bf{10}}} \right).\)

c. Compute \({\bf{P}}\left( {{\bf{6}}{\rm{ }}\# {\rm{ }}{\bf{X}}{\rm{ }}\# {\rm{ }}{\bf{10}}} \right).\)

d. Compute \({\bf{m}}\) and s2 .

e. If the store currently has in stock \({\bf{10}}\) chain-driven models and \({\bf{8}}\) shaft-driven models, what is the probability that the requests of these 15 customers can all be met from existing stock?

Suppose that trees are distributed in a forest according to a two-dimensional Poisson process with parameter\({\rm{\alpha }}\), the expected number of trees per acre, equal to\({\rm{80}}\). a. What is the probability that in a certain quarter-acre plot, there will be at most\({\rm{16}}\)trees? b. If the forest covers\({\rm{85,000}}\)acres, what is the expected number of trees in the forest? c. Suppose you select a point in the forest and construct a circle of radius\({\rm{.1}}\)mile. Let X = the number of trees within that circular region. What is the pmf of X? (Hint:\({\rm{1}}\)sq mile\({\rm{ = 640}}\)acres.)

Each of \({\rm{12}}\)refrigerators of a certain type has been returned to a distributor because of an audible, high-pitched, oscillating noise when the refrigerators are running. Suppose that \({\rm{7}}\) of these refrigerators have a defective compressor and the other \({\rm{5}}\) have less serious problems. If the refrigerators are examined in random order, let\({\rm{X}}\)be the number among the first \({\rm{6}}\) examined that have a defective compressor.

a. Calculate\({\rm{P(X = 4)}}\)and \(P(X拢 4)\)

b. Determine the probability that \({\rm{X}}\) exceeds its mean value by more than \({\rm{1}}\) standard deviation.

c. Consider a large shipment of \({\rm{400}}\)\({\rm{40}}\) refrigerators, of which 40 have defective compressors. If \({\rm{X}}\) is the number among \({\rm{15}}\) randomly selected refrigerators that have defective compressors, describe a less tedious way to calculate (at least approximately) P(X拢5)than to use the hypergeometric \({\rm{pmf}}\).

Airlines sometimes overbook flights. Suppose that for a plane with 50 seats, 55 passengers have tickets. Define the random variable Y as the number of ticketed passengers who actually show up for the flight. The probability mass function of Y appears in the accompanying table.

y

45

46

47

48

49

50

51

52

53

54

55

p(y)

.05

.10

.12

.14

.25

.17

.06

.05

.03

.02

.01

a. What is the probability that the flight will accommodateall ticketed passengers who show up?

b. What is the probability that not all ticketed passengerswho show up can be accommodated?

c. If you are the first person on the standby list (whichmeans you will be the first one to get on the plane ifthere are any seats available after all ticketed passengers have been accommodated), what is the probability that you will be able to take the flight? What is this probability if you are the third person on the standby list?

A mail-order computer business has six telephone lines. Let X denote the number of lines in use at a specified time. Suppose the pmf of X is as given in the accompanying table.

X

0

1

2

3

4

5

6

p(x)

.10

.15

.20

.25

.20

.06

.04

Calculate the probability of each of the following events.

a. {at most three lines are in use}

b. {fewer than three lines are in use}

c. {at least three lines are in use}

d. {between two and five lines, inclusive, are in use}

e. {between two and four lines, inclusive, are not in use}

f. {at least four lines are not in use}
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