Introduction to 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

Description
Introduction to Probability Models, Eleventh Edition is the latest version of Sheldon Ross's classic bestseller, used extensively by professionals and as the primary text for a first undergraduate course in applied probability. The book introduces the reader to elementary probability theory and stochastic processes, and shows how probability theory can be applied fields such as engineering, computer science, management science, the physical and social sciences, and operations research.

The hallmark features of this text have been retained in this eleventh edition: superior writing style; excellent exercises and examples covering the wide breadth of coverage of probability topic; and real-world applications in engineering, science, business and economics. The 65% new chapter material includes coverage of finite capacity queues, insurance risk models, and Markov chains, as well as updated data. The book contains compulsory material for new Exam 3 of the Society of Actuaries including several sections in the new exams. It also presents new applications of probability models in biology and new material on Point Processes, including the Hawkes process. There is a list of commonly used notations and equations, along with an instructor's solutions manual.

This text will be a helpful resource for professionals and students in actuarial science, engineering, operations research, and other fields in applied probability