# Introduction to Probability Models

## 5th Edition

**Authors:**Sheldon M. Ross

**eBook ISBN:**9781483276588

**Imprint:**Academic Press

**Published Date:**27th July 1993

**Page Count:**568

## Description

Introduction to Probability Models, Fifth Edition focuses on different probability models of natural phenomena.

This edition includes additional material in Chapters 5 and 10, such as examples relating to analyzing algorithms, minimizing highway encounters, collecting coupons, and tracking the AIDS virus. The arbitrage theorem and its relationship to the duality theorem of linear program are also covered, as well as how the arbitrage theorem leads to the Black-Scholes option pricing formula.

Other topics include the Bernoulli random variable, Chapman-Kolmogorov equations, and properties of the exponential distribution. The continuous-time Markov chains, single-server exponential queueing system, variations on Brownian motion; and variance reduction by conditioning are also elaborated.

This book is a good reference for students and researchers conducting work on probability models.

## Table of 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. Bayes' 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. Joint Probability Distribution of Functions of Random Variables

2.6. Moment Generating Functions

2.7. Limit Theorems

2.8. 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

3.6.1. 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. In Normal Sampling X- and S2 are Independent

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 Gambler's Ruin Problem

4.5.2. A Model for Algorithmic Efficiency

4.6. Branching Processes

4.7. Time Reversible Markov Chains

4.8. Markov Decision Processes

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

5.3. 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

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 Kolmogorov Differential Equations

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 325

7.5.1. Alternating Renewal Processes

7.6. Semi-Markov Processes

7.7. The Inspection Paradox

7.8. Computing the Renewal Function

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. A Shoeshine Shop

8.3.4. 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

8.7. The Model G/M/1

8.7.1. The G/M/1 Busy and Idle Periods

8.8. Multiserver Queues

8.8.1. Erlang's Loss System

8.8.2. The M/M/k Queue

8.8.3. The G/M/k Queue

8.8.4. 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. 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

10.3.1. 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 498

11.2.2. The Rejection Method

11.2.3. 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

11.7. Determining the Number of Runs

Exercises

References

Index

## Details

- No. of pages:
- 568

- Language:
- English

- Copyright:
- © Academic Press 1993

- Published:
- 27th July 1993

- Imprint:
- Academic Press

- eBook ISBN:
- 9781483276588