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应用随机过程：概率模型导论
作者：（美）Sheldon M Ross著
出版社：人民邮电出版社
出版时间：2006-03-01
ISBN：9787115145147
定价：¥88.00
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内容简介
本书实例丰富，涉及多学科各种概率模型。主要内容有随机变量、条件概率及条件期望、离散及连续马尔科夫链、指数分布、泊松过程、布朗运动及平稳过程、更新理论及排队论等，最后介绍了随机模拟。本书写得极其生动和直观，并附有大量的不同领域的习题和实用的例子。.本书可作为概率论与统计，计算机科学、保险学、物理学和社会科学、生命科学、管理科学与工程学专业随机过程基础课教材。本书是国际知名统计学家SheldonM．Ross所著的关于基础概率理论和随机过程的经典教材，被加州大学伯克利分校、哥伦比亚大学、普度大学、密歇根大学、俄勒冈州立大学、华盛顿大学等众多国外知名大学所采用。..与其他随机过程教材相比，本书非常强调实践性，内含极其丰富的例子和习题，涵盖了众多学科的各种应用；作者富于启发而又不失严密性的叙述方式，有助于读者建立概率思维方式，培养对概率理论、随机过程的直观感觉。对那些需要将概率理论应用于精算学、运筹学、物理学、工程学、计算机科学、管理学和社会科学的读者，本书是一本极好的教材或参考书。...
作者简介
SheldonM.Ross，国际知名统计学家，加州大学伯克利分校工业工程与运筹系教授。毕业于斯坦福大学统计系。研究领域包括：随机模型、仿真模拟、统计分析、金融数学等。Ross教授是多本畅销数学和统计教材的作者。
目录
1．Introduction to ProbabiIity Theory 1
1．1．IntrodUCtion 1
1．2．Sample Space and Events l
1．3．Probabilities Defined on Events 4
1．4．COIlditionaJ Probabilities 7
1．5．Independent Events 1 0
1．6．Bayes’Formula l 2
Exercises 1 5
References 2 1
2．Random VariabIes 23
2．1．Random Variables 23
2．2．Discrete Random Variables 27
2．2．1．The Bernoulli Random Variable 28
2．2．2．The Binomial Random Variable 29
2．2.3．The Geometric Random Variable 3 1
2．2．4．The POisson Random Variable 32
2.3．Continuous Random Variables 34
2.3．1．The Uniforlil Random Variable 35
2．3．2．Exponential Random Variables 36
2．3.3．Gamma Random Variables 37
2.3．4．Normal Random Variables 37
2．4．EXl9ectation Of a Ralldom Variable 38
2．4．1．The Discrete Case 38
2．4．2．The Continuous Case 4 l
2．4.3．Exlaectation of a Function Of a Random Variable 43
2．5．JOinnv Distributed Random Variables 47
2．5．1．Joint Distribution Functions 47
2．5．2．Independent Random Variables 5 l
2．5.3．Covariance and Variance of Sums of Random VariabIes 53
2．5．4．Joint Probabilitv Distribution of FUrictions of Random Variables 61
2．6．Moment Generating Functions 64
2．6．1．The Joint Distribution of the Sample Mean and Sample Variance from a Norrllal Population 74
2．7．Limit Theorems 77
2．8．StOChastic Processes 83
Exercises 85
References 96
3．ConditionaI ProbabiIity and C0nditional Expectation 97
3．1．Introduction 97
3．2．The Discrete Case 97
3．3．The Continuous Case 102
3．4．Computing Expectations by Conditioning 105
3．4．1．Computing Variances by Coilditioning 116
3．5．Computing Probabilities by Conditioning 119
3．6．Some Alaplicatioils 1 36
3．6．1．A List Model 136
3．6．2．A Random GraDh 138
3．6.3．Unifotin Priors，Polva's Urn Model，and
Bose.Einstein Statistics 146
3．6．4．Mean Time for PatteIns 150
3．6．5．A Compound Poisson Identity 1 54
3．6．6．The k.Record Values of Discrete Random Variables 158
Exercises 161
4．Markov Chains 181
4．1 Introduction 181
4．2 Chapman.Kp1mogorov Equations 185
4.3 Classifientinn nf States 189
4．4． Limiting Probabilities 200
4．5． Some Applications 2 l 3
4．5．1．The GambIer’s Ruin Problem 213
4．5．2．A MOdel fof Algorithmic Efficiency 217
4．5．3．Using a Random Walk t0 Analyze a Probabilistic Algorithm for the Satisfiabilitv Problem 220
4．6． Mean Time Spent in Transient States 226
4．7． Branching Processes 228
4．8．Time Reversible Markov Chains 232
4．9． Markov Chain Monte Carlo MethOds 243
4．1 0．Markov DecisiOn Processes 248
Exercises 252
References 268
5．The ExponentiaI Distribution and the Poisson Process 269
5．1．Introduction 269
5．2．The Exponential Distribution 270
5．2．1．Definition 270
5．2．2．Properties of t11e Exponential Distribution 272
5．2．3．Further Properties of the Exponential Distribution 279
5．2．4．ConvolutiOIlS of Exponential Random Variables 284
5．3．The Poisson Process 288
5．3．1．Cpunting Processes 288
5．3．2．Definition of t11e POisson Process 289
5．3．3．Interarrival and Waiting Time Distributions 293
5．3．4．Further Properties Of POisson Processes 295
5．3．5．Coilditional Distribution Of the Arrival Times 30l
5．3．6．Estimating Software Reliability 3 l 3
5．4．Generalizations of the Poisson Process 3 l 6
5．4．1．Nonhomogeneous Poisson Process 3 1 6
5．4．2．Compollnd Poisson Process 321
5．4．3．Conditional or Mixed Poisson Processes 327
EXeFCises 330
References 348
6．C0ntinuous.Time Markov Chains 349
6．1．Introduction 349
6．2．Continuous.Time Markov Chaias 350
6.3．Birth and Death Processes 352
6．4．The Transition Probability Function Pii(f)359
6．5．Limiting Probabilities 368
6．6．Time Reversibilitv 376
6．7．Uniformization 384
6，8．Computing the Transitioil PrObabilities 388
Exercises 390
References 399
7．RenewaI Theory and Its Applications 401
7．1． Introduction 40l
7．2． Distribution of N(f)403
7.3． Limit Theorems and Their ApplicatiOns 407
7．4． Renewal Reward Processes 416
7．5． Regenerative Processes 425
7．5．1．A1temating Renewal Processes 428
7．6． Semi.Markov Processes 434
7．7．The Inspection Paradox 437
7．8． Coml9uting the Renewal Function 440
7．9． Applications to PattelTlS 443
7．9．1．Patterns Of Discrete Random Vailables 443
7．9．2．The Expected Time t0 a Maximal Run of Distinct Values 45l
7．9．3．Increasing Runs Of Continuous Random Variables 453
7．10．The Insurance Ruin PrOblem 455
Exercises 460
RefeFences 472
8．Queueing Theory 475
8．1．IntrOdtiCtion 475
8．2．Preliminaries 476
8．2．1．Cost EquationS 477
8．2．2．Steadv.State Probabilities 478
8.3．Exl90nential Models 480
8．3．1．A Single.Server Exponential Qucueing System 480
8．3．2．A Single-Server Expoilential QacHeing System Having Finite Caloacitv 487
8．3.3．A Shoeshine Shop 490
8．3．4．A Queueing System with Bulk Service 493
8．4．Network of Oueues 496
8．4．1．Open Svstems 496
8．4．2．Closed Svstems 50l
8．5．The System M／G／1 507
8．5．1．Preliminaries：Work and Anotller COSt Identitv 507
8．5．2．Alaplication OfWork to M／G／l 508
8．5.3．BUSY Periods 509
8．6．Variations on the M／G／l 510
8．6．1．The M／G／1 with Random-Sized Batch Arrivals 510
8．6．2．Prioritv Oueues 5 l 2
8．6．3．An M／G／l optimization Example 5 I 5
8．7．The Model G／M／1 519
8．7．1．The G／M／l Busy and Idle Periods 524
8．8．A Finite Source Model 525
8．9．Multiserver 0ueues 528
8．9．1．Erlang’s Loss System 529
8．9．2．The M／M／k Oueue 530
8．9．3．The G／M／k Queue 530
8．9．4．The M／G／k Queue 532
Exercises 534
References 546
9．ReIiability Theory 547
9．1．Introduction 547
9．2．Structure Functions 547
9．2．1．Minimal Path and Minimal Cut Sets 550
9．3．Reliabilitv of Systems Of Indelaendent Comlaonents 554
9．4．Bounds on the ReliabilitV Function 559
9．4．1．Method of Inclusion and Exclusion 560
9．4．2．Second Method for Obtaining Bounds on r(p)569
9．5．System Lifle as a Function of Comoonent Lives 571
9．6．Expected System Lifetime 580
9．6．1．An Upper Bound on the Exlaected Life Of a Parallel
System 584
9．7．Systems with Repair 586
9．7．1．A Series Model with Suslaended Animation 591
Exercises 593
Refefences 600
1 0．Brownian M0tion and Stationary Processes 601
10．1．Brownian MOtion 60l
10．2．Hitting Times，Maximum Variable，and the Gambler’s Ruin
Pmhlam 605
10．3．Variations on Brownian MOtiOn 607
10．3．1．Browniall MotiOn with Drift 607
10．3．2．Geometric Brownian Motion 607
l O．4．Pricing Stock Optioas 608
1 O．4．1．An Example in Options Pricing 608
l 0．4．2．The Arbitrage Theorem 6 1 l
l O．4.3．The Black.Scholes Option Pricing Formula 6 l 4
10．5．White Noise 620
lO．6．Gaussian Processes 622
10．7．Stationarv alld Weakly Stationary Processes 625
10．8．Harlnonic Analysis Of Weaklv Stationary Processes 630
Exercises 633
References 638
1 1．SimuIation 639
11．1．Introduction 639
11．2．General Techniques for Simulating ContinUOUS Random Variables 644
11．2．1．The Inverse TransfGIrmation Method 644
1 1．2．2．The Reiection Method 645
11．2.3．The Hazard Rate Method 649
11．3．SDecial Techniques for Simulating ContinUOUS Random Variables 653
11．3．1．The Normal Distribution 653
l l.3．2．The Gamma Distribution 656
1l.3．3．The Chi.Squared Distribution 657
11．3．4．The Beta n，m)Distribution 657
ll.3．5．The Exponential Distribution..The Von Neumann Algorithm 658
11．4．SimulatinR from Discrete DistributiOns 66 l
11．4．1．The Alias Method 664
11．5．Stochastic Processes 668
11．5．1．Simulating a Nonhomogeneous Poisson Process 669
11．5．2．Simulating a Two.Dimensional POisson Process 676
11．6．Variance Reduction Techniques 679
11．6．1．Use of Anthetic Variables 680
l 1．6．2．Variance RedHetion by Conditioning 684
l 1．6．3．Control Variates 688
11．6.4．Importance Sampling 690
11．7．Determining me Number of Runs 696
11.8. Coupling from the Past 696
Exercises 699
References 707
Appendix: Solutions to Starred Exercises 709
Index 749
1．1．IntrodUCtion 1
1．2．Sample Space and Events l
1．3．Probabilities Defined on Events 4
1．4．COIlditionaJ Probabilities 7
1．5．Independent Events 1 0
1．6．Bayes’Formula l 2
Exercises 1 5
References 2 1
2．Random VariabIes 23
2．1．Random Variables 23
2．2．Discrete Random Variables 27
2．2．1．The Bernoulli Random Variable 28
2．2．2．The Binomial Random Variable 29
2．2.3．The Geometric Random Variable 3 1
2．2．4．The POisson Random Variable 32
2.3．Continuous Random Variables 34
2.3．1．The Uniforlil Random Variable 35
2．3．2．Exponential Random Variables 36
2．3.3．Gamma Random Variables 37
2.3．4．Normal Random Variables 37
2．4．EXl9ectation Of a Ralldom Variable 38
2．4．1．The Discrete Case 38
2．4．2．The Continuous Case 4 l
2．4.3．Exlaectation of a Function Of a Random Variable 43
2．5．JOinnv Distributed Random Variables 47
2．5．1．Joint Distribution Functions 47
2．5．2．Independent Random Variables 5 l
2．5.3．Covariance and Variance of Sums of Random VariabIes 53
2．5．4．Joint Probabilitv Distribution of FUrictions of Random Variables 61
2．6．Moment Generating Functions 64
2．6．1．The Joint Distribution of the Sample Mean and Sample Variance from a Norrllal Population 74
2．7．Limit Theorems 77
2．8．StOChastic Processes 83
Exercises 85
References 96
3．ConditionaI ProbabiIity and C0nditional Expectation 97
3．1．Introduction 97
3．2．The Discrete Case 97
3．3．The Continuous Case 102
3．4．Computing Expectations by Conditioning 105
3．4．1．Computing Variances by Coilditioning 116
3．5．Computing Probabilities by Conditioning 119
3．6．Some Alaplicatioils 1 36
3．6．1．A List Model 136
3．6．2．A Random GraDh 138
3．6.3．Unifotin Priors，Polva's Urn Model，and
Bose.Einstein Statistics 146
3．6．4．Mean Time for PatteIns 150
3．6．5．A Compound Poisson Identity 1 54
3．6．6．The k.Record Values of Discrete Random Variables 158
Exercises 161
4．Markov Chains 181
4．1 Introduction 181
4．2 Chapman.Kp1mogorov Equations 185
4.3 Classifientinn nf States 189
4．4． Limiting Probabilities 200
4．5． Some Applications 2 l 3
4．5．1．The GambIer’s Ruin Problem 213
4．5．2．A MOdel fof Algorithmic Efficiency 217
4．5．3．Using a Random Walk t0 Analyze a Probabilistic Algorithm for the Satisfiabilitv Problem 220
4．6． Mean Time Spent in Transient States 226
4．7． Branching Processes 228
4．8．Time Reversible Markov Chains 232
4．9． Markov Chain Monte Carlo MethOds 243
4．1 0．Markov DecisiOn Processes 248
Exercises 252
References 268
5．The ExponentiaI Distribution and the Poisson Process 269
5．1．Introduction 269
5．2．The Exponential Distribution 270
5．2．1．Definition 270
5．2．2．Properties of t11e Exponential Distribution 272
5．2．3．Further Properties of the Exponential Distribution 279
5．2．4．ConvolutiOIlS of Exponential Random Variables 284
5．3．The Poisson Process 288
5．3．1．Cpunting Processes 288
5．3．2．Definition of t11e POisson Process 289
5．3．3．Interarrival and Waiting Time Distributions 293
5．3．4．Further Properties Of POisson Processes 295
5．3．5．Coilditional Distribution Of the Arrival Times 30l
5．3．6．Estimating Software Reliability 3 l 3
5．4．Generalizations of the Poisson Process 3 l 6
5．4．1．Nonhomogeneous Poisson Process 3 1 6
5．4．2．Compollnd Poisson Process 321
5．4．3．Conditional or Mixed Poisson Processes 327
EXeFCises 330
References 348
6．C0ntinuous.Time Markov Chains 349
6．1．Introduction 349
6．2．Continuous.Time Markov Chaias 350
6.3．Birth and Death Processes 352
6．4．The Transition Probability Function Pii(f)359
6．5．Limiting Probabilities 368
6．6．Time Reversibilitv 376
6．7．Uniformization 384
6，8．Computing the Transitioil PrObabilities 388
Exercises 390
References 399
7．RenewaI Theory and Its Applications 401
7．1． Introduction 40l
7．2． Distribution of N(f)403
7.3． Limit Theorems and Their ApplicatiOns 407
7．4． Renewal Reward Processes 416
7．5． Regenerative Processes 425
7．5．1．A1temating Renewal Processes 428
7．6． Semi.Markov Processes 434
7．7．The Inspection Paradox 437
7．8． Coml9uting the Renewal Function 440
7．9． Applications to PattelTlS 443
7．9．1．Patterns Of Discrete Random Vailables 443
7．9．2．The Expected Time t0 a Maximal Run of Distinct Values 45l
7．9．3．Increasing Runs Of Continuous Random Variables 453
7．10．The Insurance Ruin PrOblem 455
Exercises 460
RefeFences 472
8．Queueing Theory 475
8．1．IntrOdtiCtion 475
8．2．Preliminaries 476
8．2．1．Cost EquationS 477
8．2．2．Steadv.State Probabilities 478
8.3．Exl90nential Models 480
8．3．1．A Single.Server Exponential Qucueing System 480
8．3．2．A Single-Server Expoilential QacHeing System Having Finite Caloacitv 487
8．3.3．A Shoeshine Shop 490
8．3．4．A Queueing System with Bulk Service 493
8．4．Network of Oueues 496
8．4．1．Open Svstems 496
8．4．2．Closed Svstems 50l
8．5．The System M／G／1 507
8．5．1．Preliminaries：Work and Anotller COSt Identitv 507
8．5．2．Alaplication OfWork to M／G／l 508
8．5.3．BUSY Periods 509
8．6．Variations on the M／G／l 510
8．6．1．The M／G／1 with Random-Sized Batch Arrivals 510
8．6．2．Prioritv Oueues 5 l 2
8．6．3．An M／G／l optimization Example 5 I 5
8．7．The Model G／M／1 519
8．7．1．The G／M／l Busy and Idle Periods 524
8．8．A Finite Source Model 525
8．9．Multiserver 0ueues 528
8．9．1．Erlang’s Loss System 529
8．9．2．The M／M／k Oueue 530
8．9．3．The G／M／k Queue 530
8．9．4．The M／G／k Queue 532
Exercises 534
References 546
9．ReIiability Theory 547
9．1．Introduction 547
9．2．Structure Functions 547
9．2．1．Minimal Path and Minimal Cut Sets 550
9．3．Reliabilitv of Systems Of Indelaendent Comlaonents 554
9．4．Bounds on the ReliabilitV Function 559
9．4．1．Method of Inclusion and Exclusion 560
9．4．2．Second Method for Obtaining Bounds on r(p)569
9．5．System Lifle as a Function of Comoonent Lives 571
9．6．Expected System Lifetime 580
9．6．1．An Upper Bound on the Exlaected Life Of a Parallel
System 584
9．7．Systems with Repair 586
9．7．1．A Series Model with Suslaended Animation 591
Exercises 593
Refefences 600
1 0．Brownian M0tion and Stationary Processes 601
10．1．Brownian MOtion 60l
10．2．Hitting Times，Maximum Variable，and the Gambler’s Ruin
Pmhlam 605
10．3．Variations on Brownian MOtiOn 607
10．3．1．Browniall MotiOn with Drift 607
10．3．2．Geometric Brownian Motion 607
l O．4．Pricing Stock Optioas 608
1 O．4．1．An Example in Options Pricing 608
l 0．4．2．The Arbitrage Theorem 6 1 l
l O．4.3．The Black.Scholes Option Pricing Formula 6 l 4
10．5．White Noise 620
lO．6．Gaussian Processes 622
10．7．Stationarv alld Weakly Stationary Processes 625
10．8．Harlnonic Analysis Of Weaklv Stationary Processes 630
Exercises 633
References 638
1 1．SimuIation 639
11．1．Introduction 639
11．2．General Techniques for Simulating ContinUOUS Random Variables 644
11．2．1．The Inverse TransfGIrmation Method 644
1 1．2．2．The Reiection Method 645
11．2.3．The Hazard Rate Method 649
11．3．SDecial Techniques for Simulating ContinUOUS Random Variables 653
11．3．1．The Normal Distribution 653
l l.3．2．The Gamma Distribution 656
1l.3．3．The Chi.Squared Distribution 657
11．3．4．The Beta n，m)Distribution 657
ll.3．5．The Exponential Distribution..The Von Neumann Algorithm 658
11．4．SimulatinR from Discrete DistributiOns 66 l
11．4．1．The Alias Method 664
11．5．Stochastic Processes 668
11．5．1．Simulating a Nonhomogeneous Poisson Process 669
11．5．2．Simulating a Two.Dimensional POisson Process 676
11．6．Variance Reduction Techniques 679
11．6．1．Use of Anthetic Variables 680
l 1．6．2．Variance RedHetion by Conditioning 684
l 1．6．3．Control Variates 688
11．6.4．Importance Sampling 690
11．7．Determining me Number of Runs 696
11.8. Coupling from the Past 696
Exercises 699
References 707
Appendix: Solutions to Starred Exercises 709
Index 749
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