Efnisyfirlit

• 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

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