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Bibliographic Details
Main Author:	Ross, Sheldon M. 1943- (Author)
Format:	Book
Language:	English
Published:	London ; San Diego, CA ; Cambridge, MA ; Oxford Elsevier, Academic Press [2019]
Edition:	Twelfth edition
Subjects:	Probabilities Stochastisches Modell Mathematisches Modell Modell Wahrscheinlichkeitsrechnung Wahrscheinlichkeitstheorie Stochastischer Prozess
Links:	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030619405&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
Physical Description:	XV, 826 Seiten Illustrationen
ISBN:	9780128143469

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adam_text	Contents Preface xi 1 Introduction to Probability Theory 1.1 Introduction 1.2 Sample Space and Events 1.3 Probabilities Defined on Events 1.4 Conditional Probabilities 1.5 Independent Events 1.6 Bayes’Formula 1.7 Probability Is a Continuous Event Function Exercises References 1 1 1 3 6 9 11 14 15 21 2 Random Variables 2.1 Random Variables 2.2 Discrete Random Variables 2.2.1 The Bernoulli Random Variable 2.2.2 The Binomial Random Variable 2.2.3 The Geometric Random Variable 2.2.4 The Poisson Random Variable 2.3 Continuous Random Variables 2.3.1 The Uniform Random Variable 2.3.2 Exponential Random Variables 2.3.3 Gamma Random Variables 2.3.4 Normal Random Variables 2.4 Expectation of a Random Variable 2.4.1 The Discrete Case 2.4.2 The Continuous Case 2.4.3 Expectation of a Function of a Random Variable 2.5 Jointly Distributed Random Variables 2.5.1 Joint Distribution Functions 2.5.2 Independent Random Variables 2.5.3 Covariance and Variance of Sums of Random Variables Properties of Covariance 2.5.4 Joint Probability Distribution of Functions of Random Variables 2.6 Moment Generating Functions 23 23 27 28 28 30 31 32 33 35 35 35 37 37 39 41 44 44 49 50 52 59 62 vi Contents 2.6.1 The Joint Distribution of the Sample Mean and Sample Variance from a Normal Population 2.7 Limit Theorems 2.8 Proof of the Strong Law of Large Numbers 2.9 Stochastic Processes Exercises References 3 4 70 73 79 84 86 99 ConditionalProbability and Conditional Expectation 3.1 Introduction 3.2 The Discrete Case 3.3 The Continuous Case 3.4 Computing Expectations by Conditioning 3.4.1 Computing Variances by Conditioning 3.5 Computing Probabilities by Conditioning 3.6 Some Applications 3.6.1 A List Model 3.6.2 A Random Graph 3.6.3 Uniform Priors, Polya’s Urn Model, and Bose-Einstein Statistics 3.6.4 Mean Time for Patterns 3.6.5 The ^-Record Values of Discrete Random Variables 3.6.6 Left Skip Free Random Walks 3.7 An Identity for Compound Random Variables 3.7.1 Poisson Compounding Distribution 3.7.2 Binomial Compounding Distribution 3.7.3 A Compounding Distribution Related to the Negative Binomial Exercises 101 101 101 104 108 120 124 143 143 145 Markov Chains 4.1 Introduction 4.2 Chapman-Kolmogorov Equations 4.3 Classification of States 4.4 Long-Run Proportions and Limiting Probabilities 4.4.1 Limiting Probabilities 4.5 Some Applications 4.5.1 The Gambler’s Ruin Problem 4.5.2 A Model for Algorithmic Efficiency 4.5.3 Using a Random Walk to Analyze a Probabilistic Algorithm for the Satisfiability Problem 4.6 Mean Time Spent in Transient States 4.7 Branching Processes 4.8 Time Reversible Markov Chains 193 193 197 205 215 232 233 233 237 152 156 159 162 168 171 172 173 174 239 245 247 251 Contents vii 4.9 Markov Chain Monte Carlo Methods 4.10 Markov Decision Processes 4.11 Hidden Markov Chains 4.11.1 Predicting the States Exercises References 261 265 269 273 275 291 5 The Exponential Distribution and thePoisson Process 5.1 Introduction 5.2 The Exponential Distribution 5.2.1 Definition 5.2.2 Properties of the Exponential Distribution 5.2.3 Further Properties of the Exponential Distribution 5.2.4 Convolutions of Exponential Random Variables 5.2.5 The Dirichlet Distribution 5.3 The Poisson Process 5.3.1 Counting Processes 5.3.2 Definition of the Poisson Process 5.3.3 Further Properties of Poisson Processes 5.3.4 Conditional Distribution of the Arrival Times 5.3.5 Estimating Software Reliability 5.4 Generalizations of the Poisson Process 5.4.1 Nonhomogeneous Poisson Process 5.4.2 Compound Poisson Process Examples of Compound Poisson Processes 5.4.3 Conditional or Mixed Poisson Processes 5.5 Random Intensity Functions and Hawkes Processes Exercises References 293 293 293 293 295 302 309 313 314 314 316 320 326 336 339 339 346 346 351 353 357 374 6 Continuous-Time Markov Chains 6.1 Introduction 6.2 Continuous-Time Markov Chains 6.3 Birth and Death Processes 6.4 The Transition Probability Function Pu Լէ) 6.5 Limiting Probabilities 6.6 Time Reversibility 6.7 The Reversed Chain 6.8 Uniformization 6.9 Computing the Transition Probabilities Exercises References 375 375 375 377 384 394 401 409 414 418 420 429 7 Renewal Theory and Its Applications 7.1 Introduction 431 431 viii Contents Distribution of N(t) Limit Theorems and Their Applications Renewal Reward Processes Regenerative Processes 7.5.1 Alternating Renewal Processes 7.6 Semi-Markov Processes 7.7 The Inspection Paradox 7.8 Computing the Renewal Function 7.9 Applications to Patterns 7.9.1 Patterns of Discrete Random Variables 7.9.2 The Expected Time to a Maximal Run of Distinct Values 7.9.3 Increasing Runs of Continuous Random Variables 7.10 The Insurance Ruin Problem Exercises References 8 7.2 7.3 7.4 7.5 432 436 450 461 464 470 473 476 479 479 486 488 489 495 506 Queueing Theory 507 8.1 Introduction 8.2 Preliminaries 8.2.1 Cost Equations 8.2.2 Steady-State Probabilities 8.3 Exponential Models 8.3.1 A Single-Server Exponential Queueing System 8.3.2 A Single-Server Exponential Queueing System Having Finite Capacity 8.3.3 Birth and Death Queueing Models 8.3.4 A Shoe Shine Shop 8.3.5 Queueing Systems with Bulk Service 8.4 Network of Queues 8.4.1 Open Systems 8.4.2 Closed Systems 8.5 The System M/G/l 8.5.1 Preliminaries: Work and Another Cost Identity 8.5.2 Application of Work to M/G/l 8.5.3 Busy Periods 8.6 Variations on the M/G/l 8.6.1 The M/G/l with Random-Sized Batch Arrivals 8.6.2 Priority Queues 8.6.3 An M/G/l Optimization Example 8.6.4 The M/G/l Queue with Server Breakdown 8.7 The Model G/M/l 8.7.! The G/M/l Busy and Idle Periods 8.8 A Finite Source Model 8.9 Multiserver Queues 507 508 508 509 512 512 522 527 534 536 540 540 544 549 549 550 552 554 554 555 558 562 565 569 570 573 Contents 8.9.1 8.9.2 8.9.3 8.9.4 Exercises ίχ Erlang’s Loss System The M/M/k Queue The G/М/к Queue The M/G/k Queue 574 575 575 577 578 9 Reliability Theory 9.1 Introduction 9.2 Structure Functions 9.2.1 Minimal Path and Minimal Cut Sets 9.3 Reliability of Systems of Independent Components 9.4 Bounds on the Reliability Function 9.4.1 Method of Inclusion and Exclusion 9.4.2 Second Method for Obtaining Bounds on r(p) 9.5 System Life as a Function of Component Lives 9.6 Expected System Lifetime 9.6.1 An Upper Bound on the Expected Life of a Parallel System 9.7 Systems with Repair 9.7.1 A Series Model with Suspended Animation Exercises References 591 591 591 594 597 601 602 610 613 620 623 625 630 632 638 10 Brownian Motion and Stationary Processes 10.1 Brownian Motion 10.2 Hitting Times, Maximum Variable, and the Gambler’s Ruin Problem 10.3 Variations on Brownian Motion 10.3.1 Brownian Motion with Drift 10.3.2 Geometric Brownian Motion 10.4 Pricing Stock Options 10.4.1 An Example in Options Pricing 10.4.2 The Arbitrage Theorem 10.4.3 The В lack-Scholes Option Pricing Formula 10.5 The Maximum of Brownian Motion with Drift 10.6 White Noise 10.7 Gaussian Processes 10.8 Stationary and Weakly Stationary Processes 10.9 Harmonic Analysis of Weakly Stationary Processes Exercises References 639 639 643 644 644 644 646 646 648 651 656 661 663 665 670 672 677 Simulation 11.1 Introduction 11.2 General Techniques for Simulating Continuous Random Variables 679 679 683 11 Contents x 11.2.1 The Inverse Transformation Method 11.2.2 The Rejection Method 11.2.3 The Hazard Rate Method 11.3 Special Techniques for Simulating Continuous Random Variables 11.3.1 The Normal Distribution 11.3.2 The Gamma Distribution 11.3.3 The Chi-Squared Distribution 11.3.4 The Beta (и, m) Distribution 11.3.5 The Exponential Distribution—The Von Neumann Algorithm 11.4 Simulating from Discrete Distributions 11.4.1 The Alias Method 11.5 Stochastic Processes 11.5.1 Simulating a Nonhomogeneous Poisson Process 11.5.2 Simulating a Two-Dimensional Poisson Process 11.6 Variance Reduction Techniques 11.6.1 Use of Antithetic Variables 11.6.2 Variance Reduction by Conditioning 11.6.3 Control Variates 11.6.4 Importance Sampling 11.7 Determining the Number of Runs 11.8 Generating from the Stationary Distribution of a Markov Chain 11.8.1 Coupling from the Past 11.8.2 Another Appro ach Exercises References 12 Coupling 12.1 A Brief Introduction 12.2 Coupling and Stochastic Order Relations 12.3 Stochastic Ordering of Stochastic Processes 12.4 Maximum Couplings, Total Variation Distance, and the Coupling Identity 12.5 Applications of the Coupling Identity 12.5.1 Applications to Markov Chains 12.6 Coupling and Stochastic Optimization 12.7 Chen-Stein Poisson Approximation Bounds Exercises 683 684 688 691 691 694 695 695 696 698 701 705 706 712 715 716 719 723 725 730 731 731 733 734 741 743 743 743 746 749 752 752 758 762 769 Solutions to Starred Exercises 773 Index 817
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title	Introduction to probability models
title_auth	Introduction to probability models
title_exact_search	Introduction to probability models
title_full	Introduction to probability models Sheldon M. Ross, University of Southern California, Los Angeles, CA, United States of America
title_fullStr	Introduction to probability models Sheldon M. Ross, University of Southern California, Los Angeles, CA, United States of America
title_full_unstemmed	Introduction to probability models Sheldon M. Ross, University of Southern California, Los Angeles, CA, United States of America
title_short	Introduction to probability models
title_sort	introduction to probability models
topic	Probabilities Stochastisches Modell (DE-588)4057633-4 gnd Mathematisches Modell (DE-588)4114528-8 gnd Modell (DE-588)4039798-1 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Stochastischer Prozess (DE-588)4057630-9 gnd
topic_facet	Probabilities Stochastisches Modell Mathematisches Modell Modell Wahrscheinlichkeitsrechnung Wahrscheinlichkeitstheorie Stochastischer Prozess
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030619405&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT rosssheldonm introductiontoprobabilitymodels

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