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• Introduction to Probability Models
• Copyright
• 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
• 7.5 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
• 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
• Back Cover