Efnisyfirlit

• 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

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