Introduction to Probability Models and Applications

An essential guide to the concepts of probability theory that puts the focus on models and applications

Introduction to Probability offers an authoritative text that presents the main ideas and concepts, as well as the theoretical background, models, and applications of probability. The authors—noted experts in the   field—include a review of problems where probabilistic models naturally arise, and discuss the methodology to tackle these problems.

A wide-range of topics are covered that include the concepts of probability and conditional probability, univariate discrete distributions, univariate continuous distributions, along with a detailed presentation of the most important probability distributions used in practice, with their main properties and applications.

Designed as a useful guide, the text contains theory of probability, de finitions, charts, examples with solutions, illustrations, self-assessment exercises, computational exercises, problems and a glossary. This important text:

• Includes classroom-tested problems and solutions to probability exercises 
• Highlights real-world exercises designed to make clear the concepts presented
• Uses Mathematica software to illustrate the text’s computer exercises
• Features applications representing worldwide situations and processes
• Offers two types of self-assessment exercises at the end of each chapter, so that students may review the material in that chapter and monitor their progress.

Written for students majoring in statistics, engineering, operations research, computer science, physics, and mathematics, Introduction to Probability: Models and Applications is an accessible text that explores the basic concepts of probability and includes detailed information on models and applications.

N. Balakrishnan, PhD, is a Distinguished University Professor in the Department of Mathematics and Statistics at McMaster University in Ontario, Canada. He is the author of over twenty Wiley books and served as co-editor of the Wiley's Encyclopedia of Statistical Sciences, Second Edition.

Markos V. Koutras, PhD, is Professor in the Department of Statistics and Insurance Science at the University of Piraeus, Greece.

Konstadinos G. Politis, PhD, is Associate Professor in the Department of Statistics and Insurance Science at the University of Piraeus, Greece.

Preface xi

1 The Concept of Probability 1

1.1 Chance Experiments – Sample Spaces 2

1.2 Operations Between Events 11

1.3 Probability as Relative Frequency 27

1.4 Axiomatic Definition of Probability 38

1.5 Properties of Probability 45

1.6 The Continuity Property of Probability 54

1.7 Basic Concepts and Formulas 60

1.8 Computational Exercises 61

1.9 Self-assessment Exercises 63

1.9.1 True–False Questions 63

1.9.2 Multiple Choice Questions 64

1.10 Review Problems 67

1.11 Applications 71

1.11.1 System Reliability 71

Key Terms 77

2 Finite Sample Spaces – Combinatorial Methods 79

2.1 Finite Sample Spaces with Events of Equal Probability 80

2.2 Main Principles of Counting 89

2.3 Permutations 96

2.4 Combinations 105

2.5 The Binomial Theorem 123

2.6 Basic Concepts and Formulas 132

2.7 Computational Exercises 133

2.8 Self-Assessment Exercises 139

2.8.1 True–False Questions 139

2.8.2 Multiple Choice Questions 140

2.9 Review Problems 143

2.10 Applications 150

2.10.1 Estimation of Population Size: Capture–Recapture Method 150

Key Terms 152

3 Conditional Probability – Independent Events 153

3.1 Conditional Probability 154

3.2 The Multiplicative Law of Probability 166

3.3 The Law of Total Probability 174

3.4 Bayes’ Formula 183

3.5 Independent Events 189

3.6 Basic Concepts and Formulas 206

3.7 Computational Exercises 207

3.8 Self-assessment Exercises 210

3.8.1 True–False Questions 210

3.8.2 Multiple Choice Questions 211

3.9 Review Problems 214

3.10 Applications 220

3.10.1 Diagnostic and Screening Tests 220

Key Terms 223

4 Discrete Random Variables and Distributions 225

4.1 Random Variables 226

4.2 Distribution Functions 232

4.3 Discrete Random Variables 247

4.4 Expectation of a Discrete Random Variable 261

4.5 Variance of a Discrete Random Variable 281

4.6 Some Results for Expectation and Variance 293

4.7 Basic Concepts and Formulas 302

4.8 Computational Exercises 303

4.9 Self-Assessment Exercises 309

4.9.1 True–False Questions 309

4.9.2 Multiple Choice Questions 310

4.10 Review Problems 313

4.11 Applications 317

4.11.1 Decision Making Under Uncertainty 317

Key Terms 320

5 Some Important Discrete Distributions 321

5.1 Bernoulli Trials and Binomial Distribution 322

5.2 Geometric and Negative Binomial Distributions 337

5.3 The Hypergeometric Distribution 358

5.4 The Poisson Distribution 371

5.5 The Poisson Process 385

5.6 Basic Concepts and Formulas 394

5.7 Computational Exercises 395

5.8 Self-Assessment Exercises 399

5.8.1 True–False Questions 399

5.8.2 Multiple Choice Questions 401

5.9 Review Problems 403

5.10 Applications 411

5.10.1 Overbooking 411

Key Terms 414

6 Continuous Random Variables 415

6.1 Density Functions 416

6.2 Distribution for a Function of a Random Variable 431

6.3 Expectation and Variance 442

6.4 Additional Useful Results for the Expectation 451

6.5 Mixed Distributions 459

6.6 Basic Concepts and Formulas 468

6.7 Computational Exercises 469

6.8 Self-Assessment Exercises 474

6.8.1 True–False Questions 474

6.8.2 Multiple Choice Questions 476

6.9 Review Problems 479

6.10 Applications 486

6.10.1 Profit Maximization 486

Key Terms 490

7 Some Important Continuous Distributions 491

7.1 The Uniform Distribution 492

7.2 The Normal Distribution 501

7.3 The Exponential Distribution 531

7.4 Other Continuous Distributions 542

7.4.1 The Gamma Distribution 543

7.4.2 The Beta Distribution 548

7.5 Basic Concepts and Formulas 555

7.6 Computational Exercises 557

7.7 Self-Assessment Exercises 561

7.7.1 True–False Questions 561

7.7.2 Multiple Choice Questions 562

7.8 Review Problems 565

7.9 Applications 573

7.9.1 Transforming Data: The Lognormal Distribution 573

Key Terms 578

Appendix A Sums and Products 579

Appendix B Distribution Function of the Standard Normal Distribution 593

Appendix C Simulation 595

Appendix D Discrete and Continuous Distributions 599

Bibliography 603

Index 605

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