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Introduction toProbability ModelsNinth EditionSheldon M. RossUniversity of CaliforniaBerkeley, CaliforniaAMSTERDAM • BOSTON • HEIDELBERG • LONDONNEW YORK • OXFORD • PARIS • SAN DIEGOSAN FRANCISCO • SINGAPORE • SYDNEY • TOKYOAcademic Press is an imprint of Elsevier
Acquisitions EditorProject ManagerMarketing ManagersCover DesignCompositionCover PrinterInterior PrinterTom SingerSarah M. HajdukLinda Beattie, Leah AckersonEric DeCiccoVTEXPhoenix ColorThe Maple-Vail Book Manufacturing GroupAcademic Press is an imprint of Elsevier30 Corporate Drive, Suite 400, Burlington, MA 01803, USA525 B Street, Suite 1900, San Diego, California 92101-4495, USA84 Theobald’s Road, London WC1X 8RR, UKThis book is printed on acid-free paper. ○∞Copyright © 2007, Elsevier Inc. All rights reserved.No part of this publication may be reproduced or transmitted in any form or by anymeans, electronic or mechanical, including photocopy, recording, or any informationstorage and retrieval system, without permission in writing from the publisher.Permissions may be sought directly from Elsevier’s Science & Technology RightsDepartment in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333,E-mail: permissions@elsevier.com. You may also complete your request on-linevia the Elsevier homepage (http://elsevier.com), by selecting “Support & Contact”then “Copyright and Permission” and then “Obtaining Permissions.”Library of Congress Cataloging-in-Publication DataApplication SubmittedBritish Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library.ISBN-13: 978-0-12-598062-3ISBN-10: 0-12-598062-0For information on all Academic Press publicationsvisit our Web site at www.books.elsevier.comPrinted in the United States of America06 07 08 09 10 9 8 7 6 5 4 3 2 1
Page 2 and 3: Introduction toProbability ModelsNi
Page 6 and 7: ContentsPreface xiii1. Introduction
Page 8 and 9: Contentsvii4. Markov Chains 1854.1.
Page 10 and 11: Contentsix8.3. Exponential Models 4
Page 12 and 13: Contentsxi11.5. Stochastic Processe
Page 14 and 15: PrefaceThis text is intended as an
Page 16 and 17: PrefacexvExamples and ExercisesMany
Page 18 and 19: Prefacexviiimportant simulation tec
Page 20 and 21: Introduction toProbability Theory11
Page 22 and 23: 1.2. Sample Space and Events 33 ′
Page 24 and 25: 1.3. Probabilities Defined on Event
Page 26 and 27: 1.4. Conditional Probabilities1.4.
Page 28 and 29: 1.4. Conditional Probabilities 9Sol
Page 30 and 31: 1.5. Independent Events 11will be q
Page 32 and 33: 1.6. Bayes’ Formula 13Solution: L
Page 34 and 35: Exercises 15Example 1.15 You know t
Page 36 and 37: Exercises 1714. The probability of
Page 38 and 39: Exercises 19*32. Suppose all n men
Page 40 and 41: References 21*48. Sixty percent of
Page 42 and 43: Random Variables22.1. Random Variab
Page 44 and 45: 2.1. Random Variables 25As a check,
Page 46 and 47: 2.2. Discrete Random Variables 27(i
Page 48 and 49: 2.2. Discrete Random Variables 29A
Page 50 and 51: 2.2. Discrete Random Variables 31wh
Page 52 and 53: 2.2. Discrete Random Variables 33No
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2.3. Continuous Random Variables 35
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2.3. Continuous Random Variables 37
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2.4. Expectation of a Random Variab
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2.5. Jointly Distributed Random Var
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= f(x)=n∑k=i− f(x)2.5. Jointly
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2.6. Moment Generating Functions 65
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2.7. Limit Theorems 772.7. Limit Th
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2.7. Limit Theorems 79Theorem 2.1 (
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2.7. Limit Theorems 81Table 2.3Area
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2.8. Stochastic Processes 83Taking
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Exercises 85order to visit i + 1 be
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Exercises 8710. Suppose three fair
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Exercises 89*27. A fair coin is ind
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Exercises 9140. Suppose that two te
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Exercises 93Suppose that E[X]=E[Y ]
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Exercises 95(i) Compute P {X = i}.(
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Conditional Probabilityand Conditio
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3.2. The Discrete Case 99Similarly,
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3.2. The Discrete Case 101Solution:
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3.3. The Continuous Case 103such th
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3.4. Computing Expectations by Cond
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3.5. Computing Probabilities by Con
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3.6. Some Applications 137The condi
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3.6. Some Applications 139To comput
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3.6. Some Applications 141Now given
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3.6. Some Applications 143Hence,∞
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3.6. Some Applications 145Figure 3.
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Hence, because there are ( nk)subse
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3.6. Some Applications 149(ii) In t
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3.6. Some Applications 151the (n+1)
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3.6. Some Applications 153sis more
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3.6. Some Applications 155to relate
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3.6. Some Applications 157Propositi
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3.7. An Identity for Compound Rando
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P r (2) ==P r (3) =r(1 − p) [α1
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Exercises 16716. The random variabl
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Exercises 169value occurs is(10 9
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Exercises 171Hint:Is it useful to k
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Exercises 173will return to its ini
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Exercises 17551. Do Exercise 50 und
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Exercises 177(b) Find the expected
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Exercises 179Figure 3.7.*73. Suppos
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Exercises 181(c) Prove the followin
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Exercises 183(a) N = min(n: X n−2
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Markov Chains44.1. IntroductionIn t
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4.1. Introduction 187If we let the
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4.2. Chapman-Kolmogorov Equations 1
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4.2. Chapman-Kolmogorov Equations 1
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4.3. Classification of States 193Su
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4.3. Classification of States 195Ex
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4.3. Classification of States 197Co
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4.3. Classification of States 199wh
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4.3. Classification of States 201α
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4.3. Classification of States 203(i
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4.4. Limiting Probabilities 205We a
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4.4. Limiting Probabilities 207Exam
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4.4. Limiting Probabilities 209Supp
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4.4. Limiting Probabilities 211Henc
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4.4. Limiting Probabilities 213the
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4.4. Limiting Probabilities 215For
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4.5. Some Applications 217orπ 2 =
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4.5. Some Applications 219and hence
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4.5. Some Applications 221M before
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4.5. Some Applications 223state n a
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4.5. Some Applications 225state i +
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4.5. Some Applications 227implying
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4.5. Some Applications 229given Boo
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4.6. Mean Time Spent in Transient S
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4.7. Branching Processes 2334.7. Br
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4.7. Branching Processes 235Therefo
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4.8. Time Reversible Markov Chains
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4.9. Markov Chain Monte Carlo Metho
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4.10. Markov Decision Processes 253
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4.10. Markov Decision Processes 255
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4.11. Hidden Markov Chains 257Examp
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4.11. Hidden Markov Chains 259Solut
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4.11. Hidden Markov Chains 261= ∑
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Exercises 263Starting withV 1 (j) =
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Exercises 2659. Suppose in Exercise
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Exercises 267Hint: Define an approp
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Exercises 269ing day with probabili
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Exercises 271(b) For any state i 1
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Exercises 27353. Find the average p
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Exercises 275*62. In Exercise 21,(a
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Exercises 277Hint:Fix i and show th
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Exercises 279(c) Let {y ja } be a s
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The ExponentialDistribution andthe
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5.2. The Exponential Distribution 2
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5.3. The Poisson Process 303(c) If
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5.3. The Poisson Process 305(ii) Th
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5.3. The Poisson Process 307Figure
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5.3. The Poisson Process 309That is
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5.3. The Poisson Process 311by cond
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5.3. The Poisson Process 313abiliti
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5.3. The Poisson Process 315Therefo
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5.3. The Poisson Process 317pendent
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5.3. The Poisson Process 319Example
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5.3. The Poisson Process 321Figure
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5.3. The Poisson Process 323As a re
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5.3. The Poisson Process 325event i
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5.3. The Poisson Process 327Therefo
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5.3. The Poisson Process 329one err
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5.4. Generalizations of the Poisson
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Exercises 347suppose that ∑ ni=1
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Exercises 34921. In a certain syste
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Exercises 35132. There are three jo
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Exercises 35344. Cars pass a certai
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Exercises 35554. A viral linear DNA
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Exercises 357Let X 1 denote the num
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Exercises 35970. For the infinite s
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Exercises 361counted events from a
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Exercises 36389. Some components of
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Continuous-TimeMarkov Chains66.1. I
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6.2. Continuous-Time Markov Chains
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6.3. Birth and Death Processes 369A
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Putting this back in terms of M(t)
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6.3. Birth and Death Processes 373(
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6.4. The Transition Probability Fun
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6.5. Limiting Probabilities 385To d
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6.5. Limiting Probabilities 387Solv
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6.5. Limiting Probabilities 389λ n
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6.5. Limiting Probabilities 391pres
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6.6. Time Reversibility 393intuitiv
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6.6. Time Reversibility 395Figure 6
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6.6. Time Reversibility 397as follo
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6.6. Time Reversibility 399Another
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6.7. Uniformization 4016.7. Uniform
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6.7. Uniformization 403state 1, it
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6.8. Computing the Transition Proba
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Exercises 407raising that matrix to
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Exercises 409max(X 1 ,...,X j ) can
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Exercises 41120. There are two mach
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Exercises 41331. A total of N custo
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References 41541. Let Y denote an e
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Renewal Theory andIts Applications7
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7.2. Distribution of N(t) 419Theref
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7.2. Distribution of N(t) 421Exampl
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7.3. Limit Theorems and Their Appli
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7.4. Renewal Reward Processes 433Th
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7.4. Renewal Reward Processes 435If
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7.4. Renewal Reward Processes 437re
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7.4. Renewal Reward Processes 439Re
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7.4. Renewal Reward Processes 441wh
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7.5. Regenerative Processes 443Rema
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7.5. Regenerative Processes 445The
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7.5. Regenerative Processes 447smal
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7.5. Regenerative Processes 449Exam
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7.5. Regenerative Processes 451Assu
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7.6. Semi-Markov Processes 453N∑
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7.7. The Inspection Paradox 4557.7.
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7.7. The Inspection Paradox 457is a
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7.8. Computing the Renewal Function
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7.9. Applications to Patterns 461Ta
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7.9. Applications to Patterns 463Th
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Case 2:In this case,The Overlap Is
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7.9. Applications to Patterns 467Re
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7.9. Applications to Patterns 469Su
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7.9. Applications to Patterns 471it
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7.10. The Insurance Ruin Problem 47
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Exercises 479Exercises1. Is it true
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Exercises 48111. A renewal process
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Exercises 483(c) Suppose that the X
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Exercises 485(b) Is the reward proc
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Exercises 487(a) Find the long run
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Exercises 489park your car in Berke
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Exercises 49151. In 1984 the countr
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QueueingTheory88.1. IntroductionIn
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8.2. Preliminaries 4958.2.1. Cost E
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8.2. Preliminaries 497Two other set
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8.3. Exponential Models 4998.3. Exp
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8.3. Exponential Models 501To deter
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8.3. Exponential Models 503where th
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8.3. Exponential Models 505amount o
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8.3. Exponential Models 507∞∑P
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8.3. Exponential Models 509balance
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8.3. Exponential Models 511Now, pot
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8.3. Exponential Models 513Figure 8
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8.3. Exponential Models 515she is s
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8.4. Network of Queues 517Note that
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8.4. Network of Queues 519To verify
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8.4. Network of Queues 521The avera
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8.4. Network of Queues 523Hence, fr
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8.4. Network of Queues 525where the
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8.4. Network of Queues 527where a j
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8.5. The System M/G/1 5298.5.2. App
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8.6. Variations on the M/G/1 531How
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8.6. Variations on the M/G/1 533Sub
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8.6. Variations on the M/G/1 535the
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8.6. Variations on the M/G/1 537var
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8.6. Variations on the M/G/1 539fro
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8.6. Variations on the M/G/1 541tom
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8.7. The Model G/M/1 543These quant
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8.7. The Model G/M/1 545(We have no
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8.7. The Model G/M/1 547rate at whi
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8.8. A Finite Source Model 5498.8.
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8.8. A Finite Source Model 551Howev
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8.9. Multiserver Queues 5538.9.1. E
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8.9. Multiserver Queues 555To deriv
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8.9. Multiserver Queues 557distribu
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Exercises 5596. Two customers move
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Exercises 561(ii) One counter is op
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Exercises 563In terms of the soluti
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Exercises 565(b) In terms of the li
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Exercises 567(a) What is the averag
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Exercises 56944. Consider the prior
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Reliability Theory99.1. Introductio
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9.2. Structure Functions 573Figure
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9.2. Structure Functions 575Example
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9.2. Structure Functions 577Figure
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9.3. Reliability of Systems of Inde
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9.3. Reliability of Systems of Inde
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9.4. Bounds on the Reliability Func
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9.5. System Life as a Function of C
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9.6. Expected System Lifetime 605an
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9.6. Expected System Lifetime 607Ma
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9.6. Expected System Lifetime 609Th
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9.7. Systems with Repair 611we have
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9.7. Systems with Repair 613∑ ni=
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9.7. Systems with Repair 615The pre
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Exercises 617Since the system fails
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Exercises 6195. Find the minimal pa
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Exercises 62113. Let r(p) be the re
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Exercises 623(a) ¯F t (a) = ¯F(t
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Brownian Motion andStationary Proce
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10.1. Brownian Motion 627the molecu
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10.2. Hitting Times, Maximum Variab
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10.3. Variations on Brownian Motion
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10.4. Pricing Stock Options 633Figu
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10.4. Pricing Stock Options 635the
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10.4. Pricing Stock Options 637Sinc
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10.4. Pricing Stock Options 639eral
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10.4. Pricing Stock Options 641wher
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10.4. Pricing Stock Options 643for
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10.5. White Noise 645we see that∫
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10.6. Gaussian Processes 647standar
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10.7. Stationary and Weakly Station
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10.8. Harmonic Analysis of Weakly S
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Exercises 657Hence, from Equation (
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Exercises 659Assume that the contin
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Exercises 66125. Compute the mean a
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Simulation1111.1. IntroductionLet X
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11.1. Introduction 665Table 11.1A R
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11.1. Introduction 667in estimating
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11.2. General Techniques for Simula
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11.3. Special Techniques for Simula
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11.4. Simulating from Discrete Dist
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11.5. Stochastic Processes 693distr
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11.5. Stochastic Processes 695Now s
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11.5. Stochastic Processes 697where
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11.5. Stochastic Processes 699To st
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11.5. Stochastic Processes 701Propo
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11.6. Variance Reduction Techniques
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11.8. Coupling from the Past 721per
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Exercises 723Exercises*1. Suppose i
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Exercises 72510. Explain how we can
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Exercises 727(e) Use part (d) to sh
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Exercises 729(a) Show that ∫ X 10
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References 731If we use simulation
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AppendixSolutions to StarredExercis
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30. (a) P {George | exactly 1 hit}=
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Chapter 2 737Chapter 24. (i) 1, 2,
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Chapter 2 73972. For the matching p
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Chapter 3 741p X|Y (3 | 3) = 1 27E[
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Chapter 3 743sive heads within the
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Chapter 4 745By symmetry,P nij = 1
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Chapter 5 74762. (a) Since π i = 1
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Chapter 5 749(d) T is the sum of n
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Chapter 6 75184. There is a record
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Chapter 6 753Then this is a birth a
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Chapter 7 75542. (a) The matrix P
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Chapter 7 757ing on the initial mac
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Chapter 8 759Chapter 82. This probl
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Chapter 8 761(b) The system goes fr
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Chapter 8 76336. The distributions
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Chapter 8 765Hence,= αμ + α E[(d
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Chapter 9 767(b) Suppose λ(t) is i
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Chapter 10 769distribution of X(t)
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Chapter 11 771Chapter 111. (a) Let
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Chapter 11 77323. Let m(t) = ∫ t0
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IndexAbsorbing state of a Markov ch
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Index 777of geometric random variab
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Index 779tables of, 68-69Monte Carl
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Index 781Poisson, 32-33, 306, 320si
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