SOLUTI ONS MANUAL FOR

SOLUTI ONS MANUAL FOR

APPLIED PROBABILITY

AND STOCHASTIC

PROCESSES

SECOND EDITION

by Frank Beichelt University of the Witwatersrand Johannesburg, South Africa 1 / 4

TABLE OF CONTENTS

CHAPTER 1: RANDOM EVENTS AND THEIR PROBABILITIES 1

CHAPTER 2: ONE-DIMNSIONAL RANDOM VARIABLES 15

CHAPTER 3: MULTIDIMENSIONAL RANDOM VARIABLES

37

CHAPTER 4: FUNCTIONS OF RANDOM VARIABLES

49

CHAPTER 5: INEQUALITIES AND LIMIT THEOREMS

57

CHAPTER 6: BASICS OF STOCHASTIC PROCESSES

65

CHAPTER 7: RANDOM POINT PROCESSES

75

CHAPTER 8: DISCRETE-TIME MARKOV CHAINS

97

CHAPTER 9: CONTINUOUS-TIME MARKOV CHAINS

113

CHAPTER 10: MARTINGALES

143

CHAPTER 11: BROWNIAN MOTION

149

CHAPTER 12: SPECTRAL ANALYSIS OF STATIONARY PROCESSES

161

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CHAPTER 1

Random Events and their Probabilities 1.1) A random experiment consists of simultaneously flipping three coins.(1) What is the corresponding sample space?

(2) Give the following events in terms of elementary events:

A = 'head appears at least two times,' B = 'head appears not more than once,' C = 'no head appears.' (3) Characterize verbally the complementary events of A, B, and C.Solution (1)

• head, 0 tail (head not). has 8 elements.

Ω={(i,j,k);i,j,k=0, 1}.Ω

(2) A={(1, 1, 0),(1, 0, 1),(0, 1, 1),(0, 1, 1)},

B={(1, 0, 0),(0, 1, 0),(0, 0, 1),(0, 0, 0)},

C={(0, 0, 0)}.

(3) 'head appears not more than once' ( ).A==B 'head appears at least two times' ( ).B==A 'at least one head appears'.C= 1.2) A random experiment consists of flipping a die to the first appearance of a '6.' What is the corresponding sample space?Solution The (countably infinite) sample space consists of all vectors with property that(z 1,z 2, ...,z k−1,z k) and all are integers between 1 and 5; .z k=6 z 1,z 2, ...,z k−1k=1, 2, ...

1.3) Castings are produced weighing either 1, 5, 10, or 20 kg. Let A, B, and C be the events that a casting weighs 1 or 5kg, exactly 10kg, and at least 10kg, respectively.Characterize verbally the events A∩B,A∪B,A∩C,and(A∪B)∩C.Solution Impossible event.A∩B A casting weighs 1, 5, or 10kg.A∪B A casting weighs 1 or 5kg.A∩C A casting weighs at least 10kg.(A∪B)∩C 1.4) Three randomly chosen persons are to be tested for the presence of gene g. Three random

events are introduced:

A = 'none of them has gene g,' B = 'at least one of them has gene g,' C = 'not more than one of them has gene g.' Determine the corresponding sample space and characterize the events A∩B,BC,andB∩C by elementary events.

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Solution Let 1 and 0 indicate whether a person has gene g or not, respectively. Then the sample space Ω consists of all the vectors with2 3 =8 (z1,z2,z3) z i= 1ifaperson has geneg, 0otherwise.is the same sample space as in exercise 1.1.Ω

A={(0, 0, 0)},B=A=Ω\A,C={(0, 0, 0),(1, 0, 0),(0, 1, 0),(0, 0, 1)}.

(impossible event),A∩B=∅ (since ),BC=B C⊂B B∩C=B∪C=A∪C=C(de Morgan rule,A⊂C).

1.5) Under which conditions are the following relations between events A and B true:

(1) (2) (3) A∩B=Ω,A∪B=Ω,A∪B=A∩B?

Solution

(1) A=B=Ω.

(2) . More generally if or A=BorB=AA⊇B A⊇B.

(3) A=B.

1.6) Visualize by a Venn diagram that the following relations between random events A, B, and C

are true:

(1) (2) (3)

A∩(B∪C)=(A∩B)∪(A∩C),(A∩B)∪(A∩B)=A,A∪B=B∪(A∩B).

1.7) (1) Verify by a Venn diagram that for three random events A, B, and C the following relation

is true:

(A\B)∩C=(A∩C)\(B∩C).

(2) Verify by the same Venn diagram that the relation is true as well.(A∩B)\C=(A\C)∩(B\C) 1.8) The random events A and B belong to a E. What events, generated by A and B,σ−algebra must belong to E (see definition 1.2)?Solution (de Morgan rules (1.1)).Ω,∅,A,A,B,B,A∪B,A∩B,A∪B=A∩B,A∩B=A∪B

Other events arise if in these events A and/or B are replaced with AandB:

A∩B=B\A,A∩B=A\B,A∪B=A\B,A∪B=B\A.

Any unions of two or more of these events do not give rise to an event, which is different from the listed ones.2

SOLUTIONS MANUAL

A B C

A∩B∩C

(A\B)∩C

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