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Chapter 5: Q.5.32 (page 214)

The time (in hours) required to repair a machine is an exponentially distributed random variable with parameters位=12. What is

(a)the probability that a repair time exceeds2hours?

(b)the conditional probability that a repair takes at least10hours, given that its duration exceeds9hours?

Short Answer

Expert verified

(a)the probability that a repair time exceeds2hours is 0.3679

(b)the conditional probability that a repair takes at least10hours, given that its duration exceeds9hours is0.6065

Step by step solution

01

Given information.

位=12

02

Explanations.

Letxbe the time required to fix a machine.

x~Exp(位=12)

pdf:f(x)=位e位x,0<xcdf:p(X<x)=1-e-位x

p(X鈮�x)=1-e(-0.5x)forx>0鈬�p(X>x)=e-位x

03

Part a Step 3 Explanation.

The probability that the repair time exceeds 2hours:

p(X>2)=e-12=0.367879

04

Part b Explanation.

p(X鈮�10|x鈮�9)=p(x>10)p(x>9)=0.0067380.011109鈮�0.6065

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Most popular questions from this chapter

If 65percent of the population of a large community is in favor of a proposed rise in school taxes, approximate the probability that a random sample of 100people will contain

(a) at least 50who are in favor of the proposition;

(b) between 60and 70inclusive who are in favor;

(c) fewer than 75in favor.

Use the result that for a nonnegative random variableY,

EY=鈭�0鈭�PY>tdt

to show that for a nonnegative random variableX,

EXn=鈭�0鈭�nxn-1PX>xdx

Hint: Start with

EXn=鈭�0鈭�PXn>tdt

and make the change of variablest=xn.

Let X and Y be independent random variables that are both equally likely to be either 1, 2, . . . ,(10)N, where N is very large. Let D denote the greatest common divisor of X and Y, and let Q k = P{D = k}.

(a) Give a heuristic argument that Q k = 1 k2 Q1. Hint: Note that in order for D to equal k, k must divide both X and Y and also X/k, and Y/k must be relatively prime. (That is, X/k, and Y/k must have a greatest common divisor equal to 1.) (b) Use part (a) to show that Q1 = P{X and Y are relatively prime} = 1 q k=1 1/k2 It is a well-known identity that !q 1 1/k2 = 蟺2/6, so Q1 = 6/蟺2. (In number theory, this is known as the Legendre theorem.) (c) Now argue that Q1 = "q i=1  P2 i 鈭� 1 P2 i  where Pi is the smallest prime greater than 1. Hint: X and Y will be relatively prime if they have no common prime factors. Hence, from part (b), we see that Problem 11 of Chapter 4 is that X and Y are relatively prime if XY has no multiple prime factors.)

You arrive at a bus stop at 10a.m., knowing that the bus will arrive at some time uniformly distributed between 10and 10:30.

(a) What is the probability that you will have to wait longer than 10minutes?

(b) If, at 10:15, the bus has not yet arrived, what is the probability that you will have to wait at least an additional 10 minutes?

(a) A fire station is to be located along a road of length A,A<鈭�. If fires occur at points uniformly chosen on localid="1646880402145" 0,A, where should the station be located so as to minimize the expected distance from the fire? That is,

choose a so as to minimize localid="1646880570154" EX-awhen X is uniformly distributed over 0,A.

(b) Now suppose that the road is of infinite length鈥� stretching from point 0outward to 鈭�. If the distance of a fire from point 0is exponentially distributed with rate 位, where should the fire station now be located? That is, we want to minimize EX-a, where X is now exponential with rate 位.
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