Introduction to probability modelsPDF|Epub|txt|kindle电子书版本网盘下载

Introduction to probability modelsPDF格式电子书版下载

注意：本站所有压缩包均有解压码： 点击下载压缩包解压工具

1Introduction to Probability Theory1

1.1Introduction1

1.2Sample Space and Events1

1.3Probabilities Defiined on Events4

1.4Conditional Probabilities6

1.5Independent Events9

1.6Bayes’ Formula11

Exercises14

References19

2Random Variables21

2.1Random Variables21

2.2Discrete Random Variables25

2.2.1 The Bernoulli Random Variable26

2.2.2 The Binomial Random Variable26

2.2.3 The Geometric Random Variable28

2.2.4 The Poisson Random Variable29

2.3Continuous Random Variables30

2.3.1 The Uniform Random Variable31

2.3.2 Exponential Random Variables32

2.3.3 Gamma Random Variables33

2.3.4 Normal Random Variables33

2.4Expectation of a Random Variable34

2.4.1 The Discrete Case34

2.4.2 The Continuous Case37

2.4.3 Expectation of a Function of a Random Variable38

2.5Jointly Distributed Random Variables42

2.5.1 Joint Distribution Functions42

2.5.2 Independent Random Variables45

2.5.3 Covariance and Variance of Sums of Random Variables46

2.5.4 Joint Probability Distribution of Functions of Random Variables55

2.6Moment Generating Functions58

2.6.1 The Joint Distribution of the Sample Mean and Sample Variance from a Normal Population66

2.7 The Distribution of the Number of Events that Occur69

2.8 Limit Theorems71

2.9 Stochastic Processes77

Exercises79

References91

3 Conditional Probability and Conditional Expectation93

3.1 Introduction93

3.2 The Discrete Case93

3.3 The Continuous Case97

3.4 Computing Expectations by Conditioning100

3.4.1 Computing Variances by Conditioning111

3.5 Computing Probabilities by Conditioning115

3.6 Some Applications133

3.6.1 A List Model133

3.6.2 A Random Graph135

3.6.3 Uniform Priors， Polya’s Urn Model， and Bose—Einstein Statistics141

3.6.4 Mean Time for Patterns146

3.6.5 The k-Record Values of Discrete Random Variables149

3.6.6 Left Skip Free Random Walks152

3.7 An Identity for Compound Random Variables157

3.7.1 Poisson Compounding Distribution160

3.7.2 Binomial Compounding Distribution161

3.7.3 A Compounding Distribution Related to the Negative Binomial162

Exercises163

4 Markov Chains183

4.1 Introduction183

4.2 Chapman-Kolmogorov Equations187

4.3 Classifiication of States194

4.4 Long-Run Proportions and Limiting Probabilities204

4.4.1 Limiting Probabilities219

4.5 Some Applications220

4.5.1 The Gambler’s Ruin Problem220

4.5.2 A Model for Algorithmic Efficiency223

4.5.3 Using a Random Walk to Analyze a Probabilistic Algorithm for the Satisfiiability Problem226

4.6 Mean Time Spent in Transient States231

4.7 Branching Processes234

4.8 Time Reversible Markov Chains237

4.9 Markov Chain Monte Carlo Methods247

4.10 Markov Decision Processes251

4.11 Hidden Markov Chains254

4.11.1 Predicting the States259

Exercises261

References275

5 The Exponential Distribution and the Poisson Process277

5.1 Introduction277

5.2 The Exponential Distribution278

5.2.1 Defiinition278

5.2.2 Properties of the Exponential Distribution280

5.2.3 Further Properties of the Exponential Distribution287

5.2.4 Convolutions of Exponential Random Variables293

5.3 The Poisson Process297

5.3.1 Counting Processes297

5.3.2 Definition of the Poisson Process298

5.3.3 Interarrival and Waiting Time Distributions301

5.3.4 Further Properties of Poisson Processes303

5.3.5 Conditional Distribution of the Arrival Times309

5.3.6 Estimating Software Reliability320

5.4 Generalizations of the Poisson Process322

5.4.1 Nonhomogeneous Poisson Process322

5.4.2 Compound Poisson Process327

5.4.3 Conditional or Mixed Poisson Processes332

5.5 Random Intensity Functions and Hawkes Processes334

Exercises338

References356

6 Continuous-Time Markov Chains357

6.1 Introduction357

6.2 Continuous-Time Markov Chains358

6.3 Birth and Death Processes359

6.4 The Transition Probability Function Pij（t）366

6.5 Limiting Probabilities374

6.6 Time Reversibility380

6.7 The Reversed Chain387

6.8 Uniformization393

6.9 Computing the Transition Probabilities396

Exercises398

References407

7 Renewal Theory and Its Applications409

7.1 Introduction409

7.2 Distribution of N（t）411

7.3 Limit Theorems and Their Applications415

7.4 Renewal Reward Processes427

7.5 Regenerative Processes436

7.5.1 Alternating Renewal Processes439

7.6 Semi-Markov Processes444

7.7 The Inspection Paradox447

7.8 Computing the Renewal Function449

7.9 Applications to Patterns452

7.9.1 Patternes of Discrete Random Variables453

7.9.2 The Expected Time to a Maximal Run of Distinct Values459

7.9.3 Increasing Runs of Continuous Random Variables461

7.10 The Insurance Ruin Problem462

Exercises468

References479

8 Queueing Theory481

8.1 Introduction481

8.2 Preliminaries482

8.2.1 Cost Equations482

8.2.2 Steady-State Probabilities484

8.3 Exponential Models486

8.3.1 A Single-Server Exponential Queueing System486

8.3.2 A Single-Server Exponential Queueing System Having Finite Capacity495

8.3.3 Birth and Death Queueing Models499

8.3.4 A Shoe Shine Shop505

8.3.5 A Queueing System with Bulk Service507

8.4 Network of Queues510

8.4.1 Open Systems510

8.4.2 Closed Systems514

8.5 The System M/G/ 1520

8.5.1 Preliminaries： Work and Another Cost Identity520

8.5.2 Application of Work to M/G/1520

8.5.3 Busy Periods522

8.6 Variations on the M/G/ 1523

8.6.1 The M/G/1 with Random-Sized Batch Arrivals523

8.6.2 Priority Queues524

8.6.3 An M/G/1 Optimization Example527

8.6.4 The M/G/l Queue with Server Breakdown531

8.7 The Model G/M/ 1534

8.7.1 The G/M/ 1 Busy and Idle Periods538

8.8 A Finite Source Model538

8.9 Multiserver Queues542

8.9.1 Erlang’s Loss System542

8.9.2 The M/M/k Queue544

8.9.3 The G/M/k Queue544

8.9.4 The M/G/k Queue546

Exercises547

References558

9 Reliability Theory559

9.1 Introduction559

9.2 Structure Functions560

9.2.1 Minimal Path and Minimal Cut Sets562

9.3 Reliability of Systems of Independent Components565

9.4 Bounds on the Reliability Function570

9.4.1 Method of Inclusion and Exclusion570

9.4.2 Second Method for Obtaining Bounds on r（p）578

9.5 System Life as a Function of Component Lives580

9.6 Expected System Lifetime587

9.6.1 An Upper Bound on the Expected Life of a Parallel System591

9.7 Systems with Repair593

9.7.1 A Series Model with Suspended Animation597

Exercises599

References606

10 Brownian Motion and Stationary Processes607

10.1 Brownian Motion607

10.2 Hitting Times， Maximum Variable， and the Gambler’s Ruin Problem611

10.3 Variations on Brownian Motion612

10.3.1 Brownian Motion with Drift612

10.3.2 Geometric Brownian Motion612

10.4 Pricing Stock Options614

10.4.1 An Example in Options Pricing614

10.4.2 The Arbitrage Theorem616

10.4.3 The Black-Scholes Option Pricing Formula619

10.5 The Maximum of Brownian Motion with Drift624

10.6 White Noise628

10.7 Gaussian Processes630

10.8 Stationary and Weakly Stationary Processes633

10.9 Harmonic Analysis of Weakly Stationary Processes637

Exercises639

References644

11 Simulation645

11.1 Introduction645

11.2 General Techniques for Simulating Continuous Random Variables649

11.2.1 The Inverse Transformation Method649

11.2.2 The Rejection Method650

11.2.3 The Hazard Rate Method654

11.3 Special Techniques for Simulating Continuous Random Variables657

11.3.1 The Normal Distribution657

11.3.2 The Gamma Distribution660

11.3.3 The Chi-Squared Distribution660

11.3.4 The Beta （n， m） Distribution661

11.3.5 The Exponential Distribution—The Von Neumann Algorithm662

11.4 Simulating from Discrete Distributions664

11.4.1 The Alias Method667

11.5 Stochastic Processes671

11.5.1 Simulating a Nonhomogeneous Poisson Process672

11.5.2 Simulating a Two-Dimensional Poisson Process677

11.6 Variance Reduction Techniques680

11.6.1 Use of Antithetic Variables681

11.6.2 Variance Reduction by Conditioning684

11.6.3 Control Variates688

11.6.4 Importance Sampling690

11.7 Determining the Number of Runs694

11.8 Generating from the Stationary Distribution of a Markov Chain695

11.8.1 Coupling from the Past695

11.8.2 Another Approach697

Exercises698

References705

Appendix： Solutions to Starred Exercises707

Index759