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• Cover image
• Title page
• Table of Contents
• Copyright
• 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
• 2.3. Continuous Random Variables
• 2.4. Expectation of a Random Variable
• 2.5. Jointly Distributed Random Variables
• 2.6. Moment Generating Functions
• 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
• 3.5. Computing Probabilities by Conditioning
• 3.6. Some Applications
• 3.7. An Identity for Compound Random Variables
• 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
• 4.5. 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
• 5: The Exponential Distribution and the Poisson Process
• 5.1. Introduction
• 5.2. The Exponential Distribution
• 5.3. The Poisson Process
• 5.4. Generalizations of the Poisson Process
• 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
• 7.5. 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
• 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
• Exercises
• 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
• 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. 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
• 11.3. 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
• 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
• 12.6. 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
• 13.4. 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