Introduction to Probability

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 xiAcknowledgments xv1 Two-Dimensional Discrete Random Variables and Distributions 11.1 Introduction 21.2 Joint Probability Function 21.3 Marginal Distributions 151.4 Expectation of a Function 241.5 Conditional Distributions and Expectations 321.6 Basic Concepts and Formulas 411.7 Computational Exercises 421.8 Self-assessment Exercises 461.8.1 True–False Questions 461.8.2 Multiple Choice Questions 471.9 Review Problems 501.10 Applications 541.10.1 Mixture Distributions and Reinsurance 54Key Terms 572 Two-Dimensional Continuous Random Variables and Distributions 592.1 Introduction 602.2 Joint Density Function 602.3 Marginal Distributions 732.4 Expectation of a Function 792.5 Conditional Distributions and Expectations 822.6 Geometric Probability 912.7 Basic Concepts and Formulas 982.8 Computational Exercises 1002.9 Self-assessment Exercises 1072.9.1 True–False Questions 1072.9.2 Multiple Choice Questions 1092.10 Review Problems 1112.11 Applications 1142.11.1 Modeling Proportions 114Key Terms 1193 Independence and Multivariate Distributions 1213.1 Introduction 1223.2 Independence 1223.3 Properties of Independent Random Variables 1373.4 Multivariate Joint Distributions 1423.5 Independence of More Than Two Variables 1563.6 Distribution of an Ordered Sample 1653.7 Basic Concepts and Formulas 1763.8 Computational Exercises 1783.9 Self-assessment Exercises 1853.9.1 True–False Questions 1853.9.2 Multiple Choice Questions 1863.10 Review Problems 1893.11 Applications 1943.11.1 Acceptance Sampling 194Key Terms 2004 Transformations of Variables 2014.1 Introduction 2024.2 Joint Distribution for Functions of Variables 2024.3 Distributions of sum, difference, product and quotient 2104.4 𝜒2, t and F Distributions 2234.5 Basic Concepts and Formulas 2364.6 Computational Exercises 2374.7 Self-assessment Exercises 2424.7.1 True–False Questions 2424.7.2 Multiple Choice Questions 2434.8 Review Problems 2464.9 Applications 2504.9.1 Random Number Generators Coverage – Planning Under Random Event Occurrences 250Key Terms 2555 Covariance and Correlation 2575.1 Introduction 2585.2 Covariance 2585.3 Correlation Coefficient 2725.4 Conditional Expectation and Variance 2815.5 Regression Curves 2935.6 Basic Concepts and Formulas 3075.7 Computational Exercises 3085.8 Self-assessment Exercises 3145.8.1 True–False Questions 3145.8.2 Multiple Choice Questions 3165.9 Review Problems 3205.10 Applications 3265.10.1 Portfolio Optimization Theory 326Key Terms 3306 Important Multivariate Distributions 3316.1 Introduction 3326.2 Multinomial Distribution 3326.3 Multivariate Hypergeometric Distribution 3446.4 Bivariate Normal Distribution 3586.5 Basic Concepts and Formulas 3716.6 Computational Exercises 3736.7 Self-Assessment Exercises 3786.7.1 True–False Questions 3786.7.2 Multiple Choice Questions 3806.8 Review Problems 3836.9 Applications 3876.9.1 The Effect of Dependence on the Distribution of the Sum 387Key Terms 3907 Generating Functions 3917.1 Introduction 3927.2 Moment Generating Function 3927.3 Moment Generating Functions of Some Important Distributions 4017.3.1 Binomial Distribution 4017.3.2 Negative Binomial Distribution 4027.3.3 Poisson Distribution 4037.3.4 Uniform Distribution 4037.3.5 Normal Distribution 4037.3.6 Gamma Distribution 4047.4 Moment Generating Functions for Sum of Variables 4077.5 Probability Generating Function 4167.6 Characteristic Function 4287.7 Generating Functions for Multivariate Case 4337.8 Basic Concepts and Formulas 4417.9 Computational Exercises 4437.10 Self-assessment Exercises 4467.10.1 True–False Questions 4467.10.2 Multiple Choice Questions 4487.11 Review Problems 4527.12 Applications 4607.12.1 Random Walks 460Key Terms 4658 Limit Theorems 4678.1 Introduction 4688.2 Laws of Large Numbers 4688.3 Central Limit Theorem 4768.4 Basic Concepts and Formulas 4928.5 Computational Exercises 4938.6 Self-assessment Exercises 4978.6.1 True–False Questions 4978.6.2 Multiple Choice Questions 4988.7 Review Problems 5018.8 Applications 5048.8.1 Use of the CLT for Capacity Planning 504Key Terms 507Appendix A Tail Probability Under Standard Normal Distribution 509Appendix B Critical Values Under Chi-Square Distribution 511Appendix C Student’s t-Distribution 515Appendix D F-Distribution: 5% (Lightface Type) and 1% (Boldface Type) Points for the F-Distribution 517Appendix E Generating Functions 521Bibliography 525Index 527