N. Balakrishnan, PhD, is Distinguished University Professor in the Department of Mathematics and Statistics at McMaster University in Ontario, Canada. He is the author of over twenty books, including 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. He is the author/coauthor/editor of 19 books (13 in Greek, 6 in English). His research interests include multivariate analysis, combinatorial distributions, theory of runs/scans/patterns, statistical quality control, and reliability theory.

Konstadinos G. Politis, PhD, is Associate Professor in the Department of Statistics and Insurance Science at the University of Piraeus. He is the author of several articles ­published in scientific journals.

Preface xi

Acknowledgments xv

1 Two-Dimensional Discrete Random Variables and Distributions 1

1.1 Introduction 2

1.2 Joint Probability Function 2

1.3 Marginal Distributions 15

1.4 Expectation of a Function 24

1.5 Conditional Distributions and Expectations 32

1.6 Basic Concepts and Formulas 41

1.7 Computational Exercises 42

1.8 Self-assessment Exercises 46

1.8.1 True-False Questions 46

1.8.2 Multiple Choice Questions 47

1.9 Review Problems 50

1.10 Applications 54

1.10.1 Mixture Distributions and Reinsurance 54

Key Terms 57

2 Two-Dimensional Continuous Random Variables and Distributions 59

2.1 Introduction 60

2.2 Joint Density Function 60

2.3 Marginal Distributions 73

2.4 Expectation of a Function 79

2.5 Conditional Distributions and Expectations 82

2.6 Geometric Probability 91

2.7 Basic Concepts and Formulas 98

2.8 Computational Exercises 100

2.9 Self-assessment Exercises 107

2.9.1 True-False Questions 107

2.9.2 Multiple Choice Questions 109

2.10 Review Problems 111

2.11 Applications 114

2.11.1 Modeling Proportions 114

Key Terms 119

3 Independence and Multivariate Distributions 121

3.1 Introduction 122

3.2 Independence 122

3.3 Properties of Independent Random Variables 137

3.4 Multivariate Joint Distributions 142

3.5 Independence of More Than Two Variables 156

3.6 Distribution of an Ordered Sample 165

3.7 Basic Concepts and Formulas 176

3.8 Computational Exercises 178

3.9 Self-assessment Exercises 185

3.9.1 True-False Questions 185

3.9.2 Multiple Choice Questions 186

3.10 Review Problems 189

3.11 Applications 194

3.11.1 Acceptance Sampling 194

Key Terms 200

4 Transformations of Variables 201

4.1 Introduction 202

4.2 Joint Distribution for Functions of Variables 202

4.3 Distributions of sum, difference, product and quotient 210

4.4 ¿2, t and F Distributions 223

4.5 Basic Concepts and Formulas 236

4.6 Computational Exercises 237

4.7 Self-assessment Exercises 242

4.7.1 True-False Questions 242

4.7.2 Multiple Choice Questions 243

4.8 Review Problems 246

4.9 Applications 250

4.9.1 Random Number Generators Coverage - Planning Under Random Event Occurrences 250

Key Terms 255

5 Covariance and Correlation 257

5.1 Introduction 258

5.2 Covariance 258

5.3 Correlation Coefficient 272

5.4 Conditional Expectation and Variance 281

5.5 Regression Curves 293

5.6 Basic Concepts and Formulas 307

5.7 Computational Exercises 308

5.8 Self-assessment Exercises 314

5.8.1 True-False Questions 314

5.8.2 Multiple Choice Questions 316

5.9 Review Problems 320

5.10 Applications 326

5.10.1 Portfolio Optimization Theory 326

Key Terms 330

6 Important Multivariate Distributions 331

6.1 Introduction 332

6.2 Multinomial Distribution 332

6.3 Multivariate Hypergeometric Distribution 344

6.4 Bivariate Normal Distribution 358

6.5 Basic Concepts and Formulas 371

6.6 Computational Exercises 373

6.7 Self-Assessment Exercises 378

6.7.1 True-False Questions 378

6.7.2 Multiple Choice Questions 380

6.8 Review Problems 383

6.9 Applications 387

6.9.1 The Effect of Dependence on the Distribution of the Sum 387

Key Terms 390

7 Generating Functions 391

7.1 Introduction 392

7.2 Moment Generating Function 392

7.3 Moment Generating Functions of Some Important Distributions 401

7.3.1 Binomial Distribution 401

7.3.2 Negative Binomial Distribution 402

7.3.3 Poisson Distribution 403

7.3.4 Uniform Distribution 403

7.3.5 Normal Distribution 403

7.3.6 Gamma Distribution 404

7.4 Moment Generating Functions for Sum of Variables 407

7.5 Probability Generating Function 416

7.6 Characteristic Function 428

7.7 Generating Functions for Multivariate Case 433

7.8 Basic Concepts and Formulas 441

7.9 Computational Exercises 443

7.10 Self-assessment Exercises 446

7.10.1 True-False Questions 446

7.10.2 Multiple Choice Questions 448

7.11 Review Problems 452

7.12 Applications 460

7.12.1 Random Walks 460

Key Terms 465

8 Limit Theorems 467

8.1 Introduction 468

8.2 Laws of Large Numbers 468

8.3 Central Limit Theorem 476

8.4 Basic Concepts and Formulas 492

8.5 Computational Exercises 493

8.6 Self-assessment Exercises 497

8.6.1 True-False Questions 497

8.6.2 Multiple Choice Questions 498

8.7 Review Problems 501

8.8 Applications 504

8.8.1 Use of the CLT for Capacity Planning 504

Key Terms 507

Appendix A Tail Probability Under Standard Normal Distribution 509

Appendix B Critical Values Under Chi-Square Distribution 511

Appendix C Student's t-Distribution 515

Appendix D F-Distribution: 5% (Lightface Type) and 1% (Boldface Type) Points for the F-Distribution 517

Appendix E Generating Functions 521

Bibliography 525

Index 527