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• Cover image
• Title page
• Table of Contents
• Copyright
• Preface
• New to This Edition
• Course
• Examples and Exercises
• Organization
• Acknowledgments
• CHAPTER 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
• Exercises
• References
• CHAPTER 2. Random Variables
• 2.1 Random Variables
• 2.2 Discrete Random Variables
• 2.2.2 The Binomial Random Variable
• 2.4 Expectation of a Random Variable
• 2.9 Stochastic process
• Exercises
• References
• CHAPTER 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
• 3.5 Computing Probabilities by Conditioning
• 3.6 Some Applications
• 3.7 An Identity for Compound Random Variables
• Exercises
• CHAPTER 4. Markov Chains
• 4.1 Introduction
• 4.2 Chapman-Kolmogorov Equations
• 4.3 Classification of States
• 4.4 Limiting Probabilities
• 4.5 Some Applications
• 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
• Exercises
• References
• CHAPTER 5. The Exponential Distribution and the Poisson Process
• 5.1 Introduction
• 5.2 The Exponential Distribution
• 5.3 The Poisson Process
• Example 5.18 (An Infinite Server Queue)
• Example 5.19 (Minimizing the Number of Encounters)
• 5.3. Proof of Proposition
• 5.4 Generalizations of the Poisson Process
• Exercises
• References
• CHAPTER 6. Continuous-Time Markov Chains
• 6.1 Introduction
• 6.2 Continuous-Time Markov Chains
• 6.3 Birth and Death Processes
• 6.6 Time Reversibility
• 6.7 Uniformization
• 6.8 Computing the Transition Probabilities
• Exercises
• References
• CHAPTER 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
• 7.5 Regenerative Processes
• 7.6 Semi−Markov Processes
• 7.7 The Inspection Paradox
• 7.8 Computing the Renewal Function
• 7.9 Applications to Patterns
• 7.10 The Insurance Ruin Problem
• Exercises
• References
• CHAPTER 8. Queueing Theory
• 8.1 Introduction
• 8.2 Preliminaries
• 8.3 Exponential Models
• 8.4 Network of Queues
• 8.5 The System M/G/1
• 8.6 Variations on the M/G/1
• 8.7 The Model G/M/1
• 8.8 A Finite Source Model
• 8.9 Multiserver Queues
• References
• CHAPTER 9. Reliability Theory
• 9.1 Introduction
• 9.2 Structure Functions
• 9.3 Reliability of Systems of Independent Components
• 9.4 Bounds on the Reliability Function
• 9.5 System Life as a Function of Component Lives
• 9.6 Expected System Lifetime
• 9.7 Systems with Repair
• Exercises
• References
• CHAPTER 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
• 10.4 Pricing Stock Options
• 10.5 White Noise
• 10.6 Gaussian Processes
• 10.7 Stationary and Weakly Stationary Processes
• 10.8 Harmonic Analysis of Weakly Stationary Processes
• Exercises
• References
• CHAPTER 11. Simulation
• 11.1 Introduction
• 11.2 General Techniques for Simulating Continuous Random Variables
• 11.3 Special Techniques for Simulating Continuous Random Variables
• 11.4 Simulating from Discrete Distributions
• 11.5 Stochastic Processes
• 11.6 Variance Reduction Techniques
• 11.7 Determining the Number of Runs
• 11.8 Generating from the Stationary Distribution of a Markov Chain
• References
• APPENDIX. Solutions to Starred Exercises
• 1 Chapter 1
• 2 Chapter 2
• 3 Chapter 3
• 4 Chapter 4
• 5 Chapter 5
• 6 Chapter 6
• 7 Chapter 7
• 8 Chapter 8
• 9 Chapter 9
• 10 Chapter 10
• 11 Chapter 11
• Index