Introduction to Probability Models, 13e

Main subject categories: • Probability • Probability models • Stochastic models • Applied probability • Interdisciplinary mathematicsIntroduction to Probability Models, Thirteenth Edition is available in two manageable volumes: an Elementary edition appropriate for undergraduate use and an Advanced edition for graduate use. Together, and through their hallmark exercises and real examples, both versions offer a comprehensive foundation of this key subject with applications across engineering, computer science, management science, the physical and social sciences and operations research. Users will find comprehensive information that introduces them to the foundations of probability modeling and stochastic processes from Random Variables, to Markov Chains and Renewal Theory.• Awarded the 2020 Textbook Excellence Award (Texty) from the Textbook and Academic Authors Association (prior edition)• Retains the useful organization that students and professors have relied on since 1972• Includes new coverage on Martingales• Offers a single source appropriate for a range of courses from undergraduate to graduate level Contents Preface New to This Edition Course Examples and Exercises Organization Acknowledgments 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 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 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 Conditional Probability 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 k-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 4 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 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 5 The Exponential Distribution and the Poisson 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 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 Pij(t) 6.5 Limiting Probabilities 6.6 Time Reversibility 6.7 The Reversed Chain 6.8 Uniformization 6.9 Computing the Transition Probabilities Exercises References 7 Renewal Theory and Its Applications 7.1 Introduction 7.2 Distribution of N(t) 7.3 Limit Theorems and Their Applications 7.4 Renewal Reward Processes 7.4.1 Renewal Reward Process Applications to Markov Chains 7.4.2 Renewal Reward Process Applications to Patterns of Markov Chain Generated Data 7.5 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 Queueing Theory 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/1 8.5.1 Preliminaries: Work and Another Cost Identity 8.5.2 Application of Work to M/G/1 8.5.3 Busy Periods 8.6 Variations on the M/G/1 8.6.1 The M/G/1 with Random-Sized Batch Arrivals 8.6.2 Priority Queues 8.6.3 An M/G/1 Optimization Example 8.6.4 The M/G/1 Queue with Server Breakdown 8.7 The Model G/M/1 8.7.1 The G/M/1 Busy and Idle Periods 8.8 A Finite Source Model 8.9 Multiserver Queues 8.9.1 Erlang’s Loss System 8.9.2 The M/M/k Queue 8.9.3 The G/M/k Queue 8.9.4 The M/G/k Queue Exercises 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 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 Black–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 11 Simulation 11.1 Introduction 11.2 General Techniques for Simulating Continuous Random Variables 11.2.1 The Inverse Transformation Method 11.2.2 The Rejection Method 11.2.3 The Hazard Rate Method Hazard Rate Method for Generating S:λs(t)=λ(t) 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 (n, 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 Method 1. Sampling a Poisson Process Method 2. Conditional Distribution of the Arrival Times Method 3. Simulating the Event Times 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 Approach 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 13 Martingales 13.1 Introduction 13.2 The Martingale Stopping Theorem 13.3 Applications of the Martingale Stopping Theorem 13.3.1 Wald’s Equation 13.3.2 Means and Variances of Pattern Occurrence Times 13.3.3 Random Walks 13.4 Submartingales Exercises Solutions to Starred Exercises Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Index Пустая страница Пустая страница "Ross's classic bestseller, Introduction to Probability Models, has been used extensively by professors as the primary text for a first undergraduate course in applied probability. It provides an Introduction to elementary probability theory and stochastic processes, and shows how probability theory can be applied to the study of phenomena in fields such as engineering, computer science, management science, the physical and social sciences, and operations research. With the addition of several new sections relating to actuaries, this text is highly recommended by the Society of Actuaries. The tenth edition contains several sections covered in the new exams."--Jacket. Approx.852 pages Awarded the 2020 Textbook Excellence Award (Texty) from the Textbook and Academic Authors Association (prior edition) Retains the useful organization that students and professors have relied on since 1972 Includes new coverage on Martingales Offers a single source appropriate for a range of courses from undergraduate to graduate level