# Introduction to Probability Models

Introduction to Probability Models, Tenth Edition, provides an introduction to elementary probability theory and stochastic processes. There are two approaches to the study of probability theory. One is heuristic and nonrigorous, and attempts to develop in students an intuitive feel for the subject that enables him or her to think probabilistically. The other approach attempts a rigorous development of probability by using the tools of measure theory. The first approach is employed in this text. The book begins by introducing basic concepts of probability theory, such as the random variable, conditional probability, and conditional expectation. This is followed by discussions of stochastic processes, including Markov chains and Poison processes. The remaining chapters cover queuing, reliability theory, Brownian motion, and simulation. Many examples are worked out throughout the text, along with exercises to be solved by students. This book will be particularly useful to those interested in learning how probability theory can be applied to the study of phenomena in fields such as engineering, computer science, management science, the physical and social sciences, and operations research. Ideally, this text would be used in a one-year course in probability models, or a one-semester course in introductory probability theory or a course in elementary stochastic processes.

Audience
Professionals and students in actuarial science, engineering, operations research, and other fields in applied probability.

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Published: December 2009

ISBN: 978-0-12-375686-2

## Reviews

• Praise from Reviewers: âI think Ross has done an admirable job of covering the breadth of applied probability. Ross writes fantastic problems which really force the students to think divergently...The examples, like the exercises are great.â - Matt Carlton, California Polytechnic Institute âThis is a fascinating introduction to applications from a variety of disciplines. Any curious student will love this book." - Jean LeMaire, University of Pennsylvania âThis book may be a model in the organization of the education process. I would definitely rate this text to be the best probability models book at its level of difficulty...far more sophisticated and deliberate than its competitors.â - Kris Ostaszewski, University of Illinois

## Contents

• Preface

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 Bayesâ Formula

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

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 The Distribution of the Number of Events that Occur

2.8 Limit Theorems

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, Polyas 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 Limiting Probabilities

4.5 Some Applications

4.5.1 The Gamblers 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

5.3 The Poisson Process

5.3.1 Counting Processes

5.3.2 Definition of the Poisson Process

5.3.3 Interarrival and Waiting Time Distributions

5.3.4 Further Properties of Poisson Processes

5.3.5 Conditional Distribution of the Arrival Times

5.3.6 Estimating Software Reliability

5.4 Generalizations of the Poisson Process

5.4.1 Nonhomogeneous Poisson Process

5.4.2 Compound Poisson Process

5.4.3 Conditional or Mixed Poisson 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 Uniformization

6.8 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

7.8 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

8.3 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 A Queueing System 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 Erlangs 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

References

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

9.6.1 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 White Noise

10.6 Gaussian Processes

10.7 Stationary and Weakly Stationary Processes

10.8 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

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

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

Appendix: Solutions to Starred Exercises

Index