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TitleFundamentals of Applied Probability and Random Processes Oliver C Ibe Solution
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Table of Contents
                            Chapt1
	Chapter 1 Basic Probability Concepts
		1.1 Let X denote the outcome of the first roll of the die and Y the outcome of the second roll. Then (x, y) denotes the event {X = x and Y = y}.
		1.2 Let (a, b) denote the event {A = a and B = b}.
		1.3 We represent the outcomes of the experiment as a two-dimensional diagram where the horizontal axis is the outcome of the fir...
		1.4 We represent the outcomes of the experiment as a two-dimensional diagram where the horizontal axis is the outcome of the fir...
		1.5 The experiment can stop after the first trial if the outcome is a head (H). If the first trial results in a tail (T), we try...
		1.6 The sample space for the experiment is as shown below:
		1.7 Let B denote the event that Bob wins the game, C the event that Chuck wins the game, and D the event that Dan wins the game. Then denote the complements of B, C, and D, respectively. The sample space for this game is the following:
		1.8 Let M denote the number of literate males in millions and F the number of literate females in millions. Then we know that
		1.9 We are given that A and B are two independent events with and . Now,
		1.10 Recall that denotes the probability that either A occurs or B occurs or both events occur. Thus, if Z is the event that exactly one of the two events occurs, then
		1.11 We are given two events A and B with , , and .
		1.12 We are given two events A and B with , , and . We know that
		1.13 We are given two events A and B with , , and . From the De Morgan’s first law we have that
		1.14 We are given two events A and B with . , and .
		1.15 We use the tree diagram to solve the problem. First, we note that there are two cases to consider in this problem:
		1.16 We are given a box that contains 9 red balls, 6 white balls, and 5 blue balls from which 3 balls are drawn successively. The total number of balls is 20.
		1.17 Given that A is the set of positive even integers, B the set of positive integers that are divisible by 3, and C the set of positive odd integers. Then the following events can be described as follows:
		1.18 Given a box that contains 4 red balls labeled , , , and ; and 3 white balls labeled , , and . If a ball is randomly drawn from the box, then
		1.19 Given a box that contains 50 computer chips of which 8 are known to be bad. A chip is selected at random and tested.
		1.20 S = {A, B, C, D}. Thus, all the possible subsets of S are as follows:
		1.21 Let S denote the universal set, and assume that the three sets A, B, and C have intersections as shown below.
		1.22 Given: S = {2, 4, 6, 8, 10, 12, 14}, A = {2, 4, 8}, and B = {4, 6, 8, 12}.
		1.23 denotes the event that switch is closed, k = 1, 2, 3, 4, and denotes the event that there is a closed path between nodes A and B. Then is given as follows:
		1.24 A, B, and C are three given events.
		1.25 Let denote the probability that Mark attends class on a given day and the probability that Lisa attends class on a given day. Then we know that and . Let M denote the event that Mark attends class and L the event that Lisa attends class.
		1.26 Let R denote the event that it rains and CF the event that the forecast is correct. We can use the following diagram to solve the problem:
		1.27 Let m denote the fraction of adult males that are unemployed and f the fraction of adult females that are unemployed. We use the following tree diagram to solve the problem.
		1.28 If three companies are randomly selected from the 100 companies without replacement, the probability that each of the three has installed WLAN is given by
		1.29 Let A denote the event that a randomly selected car is produced in factory A, and B the event that it is produced in factory B. Let D denote the event that a randomly selected car is defective. Now we are given the following:
		1.30 Let X denote the event that there is at least one 6, and Y the event that the sum is at least 9. Then X can be represented as follows.
		1.31 Let F denote the event that Chuck is a fool and T the event that he is a thief. Then we know that , , and . From the De Moragn’s first law,
		1.32 Let M denote the event that a married man votes and W the event that a married woman votes. Then we know that
		1.33 Let L denote the event that the plane is late and R the event that the forecast calls for rain. Then we know that
		1.34 We are given the following communication channel with the input symbol set and the output symbol set as well as the transition (or conditional) probabilities defined by that are indicated on the directed links. Also, .
		1.35 Let M denote the event that a student is a man, W the event that a student is a woman, and F the event that a student is a ...
		1.36 Let J denote the event that Joe is innocent, C the event that Chris testifies that Joe is innocent, D the event that Dana t...
		1.37 Let A denote the event that a car is a brand A car, B the event that it is a brand B car, C the event that it is a brand C car, and R the event that it needs a major repair during the first year of purchase. We know that
		1.38 The sample space of the experiment is S = {HH, HT, TH, TT}. Let A denote the event that at least one coin resulted in heads...
		1.39 The defined events are A ={The first die is odd}, B = {The second die is odd}, and C = {The sum is odd}. The sample space for the experiment is as follows:
		1.40 We are given a game that consists of two successive trials in which the first trial has outcome A or B and the second trial has outcome C or D. The probabilities of the four possible outcomes of the game are as follows:
		1.41 Since the two events A and B are mutually exclusive, we have that
		1.42 We are given 4 married couples who bought 8 seats in a row for a football game.
		1.43 We have that a committee consisting of 3 electrical engineers and 3 mechanical engineers is to be formed from a group of 7 electrical engineers and 5 mechanical engineers.
		1.44 The Stirling’s formula is given by .
		1.45 The number of different committees that can be formed is
		1.46 The number of ways of indiscriminately choosing two from the 50 states is
		1.47 We are required to form a committee of 7 people from 10 men and 12 women.
		1.48 Five departments labeled A, B, C, D, and E, send 3 delegates each to the college’s convention for a total of 15 delegates. A committee of 4 delegates is formed.
		1.49 We are given the system shown below, where the number against each component indicates the probability that the component w...
		1.50 In the structure shown the reliability functions of the switches , , , and are , , , and , respectively, and the switches are assumed to fail independently.
		1.51 The switches labeled that interconnect nodes A and B have the reliability functions , respectively, and are assumed to fail independently.
		1.52 We are given the bridge network that interconnects nodes A and B. The switches labeled have the reliability functions , respectively, and are assumed to fail independently.
		1.53 We are given the following network that interconnects nodes A and B, where the switches labeled have the reliability functions , respectively, and fail independently.
Chapt2
	Chapter 2 Random Variables
Chapt3
	Chapter 3 Moments of Random Variables
Chapt4
	CHAPTER 4 Special Probability Distributions
Chapt5
	Chapter 5 Multiple Random Variables
Chapt6
	Chapter 6 Functions of Random Variables
Chapt7
	Chapter 7 Transform Methods
Chapt8
	Chapter 8 Introduction to Random Processes
Chapt9
	Chapter 9 Linear Systems with Random Inputs
Chapt10
	Chapter 10 Some Models of Random Processes
Chapt11
	Chapter 11 Introduction to Statistics
                        
Document Text Contents
Page 1

Chapter 1 Basic Probability Concepts

Section 1.2. Sample Space and Events

1.1 Let X denote the outcome of the first roll of the die and Y the outcome of the second
roll. Then (x, y) denotes the event {X = x and Y = y}.
a. Let U denote the event that the second number is twice the first; that is, . Then

U can be represented by

Since there are 36 equally likely sample points in the experiment, the probability of
U is given by

b. Let V denote the event that the second number is greater than the first. Then V can
be represented by

Thus, the probability of V is given by

and the probability q that the second number is not greater than the first is given by

c. Let W denote the event that at least one number is greater than 3. If we use “na” to
denote that an entry is not applicable, then W can be represented by

y 2x=

U 1 2,( ) 2 4,( ) 3 6,( ), ,{ }=

P U[ ] 3 36⁄ 1 12⁄= =

V 1 2,( ) 1 3,( ) 1 4,( ) 1 5,( ) 1 6,( ) 2 3,( ) 2 4,( ) 2 5,( ) 2 6,( ) 3 4,( ) 3.5( ) 3 6,( ) 4.5( ) 4.6( ) 5.6( ), , , , , , , , , , , , , ,{ }=

P V[ ] 15 36⁄ 5 12⁄= =

q 1 P V[ ]– 7 12⁄= =
Fundamentals of Applied Probability and Random Processes 1

Page 2

Basic Probability Concepts
Thus, the probability of W is given by

1.2 Let (a, b) denote the event {A = a and B = b}.
(a) Let X denote the event that at least one 4 appears. Then X can be represented by

Thus, the probability of X is given by

(b) Let Y denote the event that just one 4 appears. Then Y can be represented by

Thus, the probability of Y is given by

(c) Let Z denote the event that the sum of the face values is 7. Then Z can be represented
by

W

na na na 1 4,( ) 1 5,( ) 1 6,( )
na na na 2 4,( ) 2 5,( ) 2 6,( )
na na na 3 4,( ) 3 5,( ) 3 6,( )
4 1,( ) 4 2,( ) 4 3,( ) 4 4,( ) 4 5,( ) 4 6,( )
5 1,( ) 5 2,( ) 5 3,( ) 5 4,( ) 5 5,( ) 5 6,( )
6 1,( ) 6 2,( ) 6 3,( ) 6 4,( ) 6 5,( ) 6 6,( )

 
 
 
 
 
 
 
 
 
 
 

=

P W[ ] 27 36⁄ 3 4⁄= =

X 1 4,( ) 2 4,( ) 3 4,( ) 4 4,( ) 5 4,( ) 6 4,( ) 4 1,( ) 4 2,( ) 4 3,( ) 4 5,( ) 4 6,( ), , , , , , , , , ,{ }=

P X[ ] 11 36⁄=

Y 1 4,( ) 2 4,( ) 3 4,( ) 5 4,( ) 6 4,( ) 4 1,( ) 4 2,( ) 4 3,( ) 4 5,( ) 4 6,( ), , , , , , , , ,{ }=

P X[ ] 10 36⁄ 5 18⁄= =

Z 1 6,( ) 2 5,( ) 3 4,( ) 4 3,( ) 5 2,( ) 6 1,( ), , , , ,{ }=
2 Fundamentals of Applied Probability and Random Processes

Page 177

Thus, the expected value of V is given by

6.17 The system arrangement of A and B is as shown below.

Let the random variable U denote the lifetime of A, and let the random variable V denote
the lifetime of B. Then we know that the PDFs of U and V are given by

The time, X, until the system fails is given by X = min(U, V). If we assume that A and B
fail independently, then U and V are independent and the PDF of X can be obtained as
follows:

pV v( )

11 36⁄ v 1=
9 36⁄ v 2=
7 36⁄ v 3=
5 36⁄ v 4=
3 36⁄ v 5=
1 36⁄ v 6=










=

E V[ ] 1
36
------ 1 11( ) 2 9( ) 3 7( ) 4 5( ) 5 3( ) 6 1( )+ + + + +{ } 91

36
------= =

A B

λ µ

fU u( ) λe
λu–= E U[ ] 1

λ
--- 200 λ⇒ 1

200
--------- u 0≥,= = =

fV v( ) µe
µu–= E V[ ] 1

µ
--- 400 µ⇒ 1

400
--------- v 0≥,= = =
Fundamentals of Applied Probability and Random Processes 177

Page 178

Functions of Random Variables
Since and , we obtain

6.18 The system arrangement of components A and B is as shown below.

Let the random variable U denote the lifetime of A, and let the random variable V denote
the lifetime of B. Then we know that the PDFs of U and V are given by

FX x( ) P X x≤[ ] P min U V,( ) x≤[ ] P U x≤( ) V x≤( )∪[ ]= = =

P U x≤[ ] P V x≤[ ] P U x V x≤,≤[ ]–+ FU x( ) FV x( ) FUV x x,( )–+==

FU x( ) FV x( ) FU x( )FV x( )–+=

fX x( ) xd
d FX x( ) fU x( ) fV x( ) fU x( )FV x( ) FU x( )fV x( )––+= =

fU x( ) 1 FV x( )–{ } fV x( ) 1 FU x( )–{ }+=

FU x( ) 1 e
λx––= FV x( ) 1 e

µx––=

fX x( ) λe
λx– e µx– µe µx– e λx–+ λ µ+( )e λ µ+( )x–= =

1
200
--------- 1

400
---------+ 

  e
1

200
--------- 1

400
---------+ 

  x– 3
400
---------e 3x 400⁄( )–== x 0≥

A

B

λ

µ

fU u( ) λe
λu–= E U[ ] 1

λ
--- 200 λ⇒ 1

200
--------- u 0≥,= = =

fV v( ) µe
µu–= E V[ ] 1

µ
--- 400 µ⇒ 1

400
--------- v 0≥,= = =
178 Fundamentals of Applied Probability and Random Processes

Page 353

The values of a and b that make the line best fit the above data are given by

Thus, the best line is .

3 4 9 12 16

3 3 9 9 9

0 0 0 0 0

2 4 4 8 16

2 4 4 8 16

3 3 9 9 9

0 0 0 0 0

3 4 9 12 16

2 3 4 6 9

1 1 1 1 1

3 3 9 9 9

2 1 4 2 1

x y x2 xy y2

x∑ 24= y∑ 30= x
2

∑ 62= xy∑ 76= y
2

∑ 102=

b∗

n xiyi
i 1=

n

∑ xi
i 1=

n

∑ yi
i 1=

n

∑–

n xi
2

i 1=

n

∑ xi
i 1=

n


 
 
 
 

2



----------------------------------------------------- 12 76( ) 24 30( )–
12 62( ) 24( )2–

---------------------------------------- 912 720–
744 576–
------------------------ 1.143= = = =

a∗

yi
i 1=

n

∑ b xi
i 1=

n

∑–

n
----------------------------------- 30 1.143 24( )–

12
------------------------------------ 0.214= = =

y 0.214 1.143x+=
Fundamentals of Applied Probability and Random Processes 353

Page 354

Introduction to Statistics
b. The estimate for the number of children that a coupled who had planned to have 5
children actually had is given by .y 0.214 5 1.143( )+ 5.929 6= = =
354 Fundamentals of Applied Probability and Random Processes

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