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1 Introduction to Probability Theory 1
1£®1 Introduction 1
1£®2 Sample Space and Events 1
1£®3 Probabilities Defined on Events 3
1£®4 Conditional Probabilities 6
1£®5 Independent Events 9
1£®6 Bayes¡¯ Formula 11
1£®7 Probability Is a Continuous Event Function 14
Exercises 15
References 21
2 Random Variables 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 28
2£®2£®3 The Geometric Random Variable 30
2£®2£®4 The Poisson Random Variable 31
2£®3 Continuous Random Variables 32
2£®3£®1 The Uniform Random Variable 33
2£®3£®2 Exponential Random Variables 35
2£®3£®3 Gamma Random Variables 35
2£®3£®4 Normal Random Variables 35
2£®4 Expectation of a Random Variable 37
2£®4£®1 The Discrete Case 37
2£®4£®2 The Continuous Case 39
2£®4£®3 Expectation of a Function of a Random Variable 41
2£®5 Jointly Distributed Random Variables 44
2£®5£®1 Joint Distribution Functions 44
2£®5£®2 Independent Random Variables 49
2£®5£®3 Covariance and Variance of Sums of Random Variables 50 Properties of Covariance 52
2£®5£®4 Joint Probability Distribution of Functions of Random Variables 59
2£®6 Moment Generating Functions 62
2£®6£®1 The Joint Distribution of the Sample Mean and Sample Variance from a Normal Population 70
2£®7 Limit Theorems 73
2£®8 Proof of the Strong Law of Large Numbers 79
2£®9 Stochastic Processes 84
Exercises 86
References 99
3 Conditional Probability and Conditional Expectation 101
3£®1 Introduction 101
3£®2 The Discrete Case 101
3£®3 The Continuous Case 104
3£®4 Computing Expectations by Conditioning 108
3£®4£®1 Computing Variances by Conditioning 120
3£®5 Computing Probabilities by Conditioning 124
3£®6 Some Applications 143
3£®6£®1 A List Model 143
3£®6£®2 A Random Graph 145
3£®6£®3 Uniform Priors£¬ Polya¡¯s Urn Model£¬ and Bose¨CEinstein Statistics 152
3£®6£®4 Mean Time for Patterns 156
3£®6£®5 The k-Record Values of Discrete Random Variables 159
3£®6£®6 Left Skip Free Random Walks 162
3£®7 An Identity for Compound Random Variables 168
3£®7£®1 Poisson Compounding Distribution 171
3£®7£®2 Binomial Compounding Distribution 172
3£®7£®3 A Compounding Distribution Related to the Negative Binomial 173
Exercises 174
4 Markov Chains 193
4£®1 Introduction 193
4£®2 Chapman¨CKolmogorov Equations 197
4£®3 Classification of States 205
4£®4 Long-Run Proportions and Limiting Probabilities 215
4£®4£®1 Limiting Probabilities 232
4£®5 Some Applications 233
4£®5£®1 The Gambler¡¯s Ruin Problem 233
4£®5£®2 A Model for Algorithmic Efficiency 237
4£®5£®3 Using a Random Walk to Analyze a Probabilistic Algorithm for the Satisfiability Problem 239
4£®6 Mean Time Spent in Transient States 245
4£®7 Branching Processes 2475£®2£®2 Properties of the Exponential Distribution 295
4£®8 Time Reversible Markov Chains 251
4£®9 Markov Chain Monte Carlo Methods 261
4£®10 Markov Decision Processes 265
4£®11 Hidden Markov Chains 269
4£®11£®1 Predicting the States 273
Exercises 275
References 291
5 The Exponential Distribution and the Poisson Process 293
5£®1 Introduction 293
5£®2 The Exponential Distribution 293
5£®2£®1 Definition 293
5£®2£®2 Properties of the Exponential Distribution 295
5£®2£®3 Further Properties of the Exponential Distribution 302
5£®2£®4 Convolutions of Exponential Random Variables 309
5£®2£®5 The Dirichlet Distribution 313
5£®3 The Poisson Process 314
5£®3£®1 Counting Processes 314
5£®3£®2 Definition of the Poisson Process 316
5£®3£®3 Further Properties of Poisson Processes 320
5£®3£®4 Conditional Distribution of the Arrival Times 326
5£®3£®5 Estimating Software Reliability 336
5£®4 Generalizations of the Poisson Process 339
5£®4£®1 Nonhomogeneous Poisson Process 339
5£®4£®2 Compound Poisson Process 346
Examples of Compound Poisson Processes 346
5£®4£®3 Conditional or Mixed Poisson Processes 351
5£®5 Random Intensity Functions and Hawkes Processes 353
Exercises 357
References 374
6 Continuous-Time Markov Chains 375
6£®1 Introduction 375
6£®2 Continuous-Time Markov Chains 375
6£®3 Birth and Death Processes 377
6£®4 The Transition Probability Function Pi j (t) 384
6£®5 Limiting Probabilities 394
6£®6 Time Reversibility 401
6£®7 The Reversed Chain 409
6£®8 Uniformization 414
6£®9 Computing the Transition Probabilities 418
Exercises 420
References 429
7 Renewal Theory and Its Applications 431
7£®1 Introduction 431
7£®2 Distribution of N (t) 432
7£®3 Limit Theorems and Their Applications 436
7£®4 Renewal Reward Processes 450
7£®5 Regenerative Processes 461
7£®5£®1 Alternating Renewal Processes 464
7£®6 Semi-Markov Processes 470
7£®7 The Inspection Paradox 473
7£®8 Computing the Renewal Function 476
7£®9 Applications to Patterns 479
7£®9£®1 Patterns of Discrete Random Variables 479
7£®9£®2 The Expected Time to a Maximal Run of Distinct Values 486
7£®9£®3 Increasing Runs of Continuous Random Variables 488
7£®10 The Insurance Ruin Problem 489
Exercises 495
References 506
8 Queueing Theory 507
8£®1 Introduction 507
8£®2 Preliminaries 508
8£®2£®1 Cost Equations 508
8£®2£®2 Steady-State Probabilities 509
8£®3 Exponential Models 512
8£®3£®1 A Single-Server Exponential Queueing System 512
8£®3£®2 A Single-Server Exponential Queueing System Having Finite Capacity 522
8£®3£®3 Birth and Death Queueing Models 527
8£®3£®4 A Shoe Shine Shop 534
8£®3£®5 Queueing Systems with Bulk Service 536
8£®4 Network of Queues 540
8£®4£®1 Open Systems 540
8£®4£®2 Closed Systems 544
8£®5 The System M G 1 549
8£®5£®1 Preliminaries£º Work and Another Cost Identity 549
8£®5£®2 Application of Work to M G 1 550
8£®5£®3 Busy Periods 552
8£®6 Variations on the M G 1 554
8£®6£®1 The M G 1 with Random-Sized Batch Arrivals 554
8£®6£®2 Priority Queues 555
8£®6£®3 An M G 1 Optimization Example 558
8£®6£®4 The M G 1 Queue with Server Breakdown 562
8£®7 The Model G M 1 565
8£®7£®1 The G M 1 Busy and Idle Periods 569
8£®8 A Finite Source Model 570
8£®9 Multiserver Queues 573
8£®9£®1 Erlang¡¯s Loss System 574
8£®9£®2 The M M k Queue 575
8£®9£®3 The G M k Queue 575
8£®9£®4 The M G k Queue 577
Exercises 578
9 Reliability Theory 591
9£®1 Introduction 591
9£®2 Structure Functions 591
9£®2£®1 Minimal Path and Minimal Cut Sets 594
9£®3 Reliability of Systems of Independent Components 597
9£®4 Bounds on the Reliability Function 601
9£®4£®1 Method of Inclusion and Exclusion 602
9£®4£®2 Second Method for Obtaining Bounds on r(p) 610
9£®5 System Life as a Function of Component Lives 613
9£®6 Expected System Lifetime 620
9£®6£®1 An Upper Bound on the Expected Life of a Parallel System 623
9£®7 Systems with Repair 625
9£®7£®1 A Series Model with Suspended Animation 630
Exercises 632
References 638
10 Brownian Motion and Stationary Processes 639
10£®1 Brownian Motion 639
10£®2 Hitting Times£¬ Maximum Variable£¬ and the Gambler¡¯s Ruin Problem 643
10£®3 Variations on Brownian Motion 644
10£®3£®1 Brownian Motion with Drift 644
10£®3£®2 Geometric Brownian Motion 644
10£®4 Pricing Stock Options 646
10£®4£®1 An Example in Options Pricing 646
10£®4£®2 The Arbitrage Theorem 648
10£®4£®3 The Black¨CScholes Option Pricing Formula 651
10£®5 The Maximum of Brownian Motion with Drift 656
10£®6 White Noise 661
10£®7 Gaussian Processes 663
10£®8 Stationary and Weakly Stationary Processes 665
10£®9 Harmonic Analysis of Weakly Stationary Processes 670
Exercises 672
References 677
11 Simulation 679
11£®1 Introduction 679
11£®2 General Techniques for Simulating Continuous Random Variables 683
11£®2£®1 The Inverse Transformation Method 683
11£®2£®2 The Rejection Method 684
11£®2£®3 The Hazard Rate Method 688
11£®3 Special Techniques for Simulating Continuous Random Variables 691
11£®3£®1 The Normal Distribution 691
11£®3£®2 The Gamma Distribution 694
11£®3£®3 The Chi-Squared Distribution 695
11£®3£®4 The Beta (n£¬ m) Distribution 695
11£®3£®5 The Exponential Distribution¡ªThe Von Neumann Algorithm 696
11£®4 Simulating from Discrete Distributions 698
11£®4£®1 The Alias Method 701
11£®5 Stochastic Processes 705
11£®5£®1 Simulating a Nonhomogeneous Poisson Process 706
11£®5£®2 Simulating a Two-Dimensional Poisson Process 712
11£®6 Variance Reduction Techniques 715
11£®6£®1 Use of Antithetic Variables 716
11£®6£®2 Variance Reduction by Conditioning 719
11£®6£®3 Control Variates 723
11£®6£®4 Importance Sampling 725
11£®7 Determining the Number of Runs 730
11£®8 Generating from the Stationary Distribution of a Markov Chain 731
11£®8£®1 Coupling from the Past 731
11£®8£®2 Another Approach 733
Exercises 734
References 741
12 Coupling 743
12£®1 A Brief Introduction 743
12£®2 Coupling and Stochastic Order Relations 743
12£®3 Stochastic Ordering of Stochastic Processes 746
12£®4 Maximum Couplings£¬ Total Variation Distance£¬ and the Coupling Identity 749
12£®5 Applications of the Coupling Identity 752
12£®5£®1 Applications to Markov Chains 752
12£®6 Coupling and Stochastic Optimization 758
12£®7 Chen¨CStein Poisson Approximation Bounds 762
Exercises 769
Solutions to Starred Exercises 773
Index 817