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Index
Cover image
Title page
Table of Contents
Copyright
Preface
New to This Edition
Course
Examples and Exercises
Organization
Acknowledgments
Introduction to Probability Theory
Abstract
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
Random Variables
Abstract
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 The Distribution of the Number of Events that Occur
2.8 Limit Theorems
2.9 Stochastic Processes
Exercises
References
Conditional Probability and Conditional Expectation
Abstract
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
Markov Chains
Abstract
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
The Exponential Distribution and the Poisson Process
Abstract
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
Continuous-Time Markov Chains
Abstract
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
Renewal Theory and Its Applications
Abstract
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
Queueing Theory
Abstract
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
References
Reliability Theory
Abstract
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
Brownian Motion and Stationary Processes
Abstract
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
Simulation
Abstract
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
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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