Introduction to Probability

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 xi1 The Concept of Probability 11.1 Chance Experiments – Sample Spaces 21.2 Operations Between Events 111.3 Probability as Relative Frequency 271.4 Axiomatic Definition of Probability 381.5 Properties of Probability 451.6 The Continuity Property of Probability 541.7 Basic Concepts and Formulas 601.8 Computational Exercises 611.9 Self-assessment Exercises 631.9.1 True–False Questions 631.9.2 Multiple Choice Questions 641.10 Review Problems 671.11 Applications 711.11.1 System Reliability 71Key Terms 772 Finite Sample Spaces – Combinatorial Methods 792.1 Finite Sample Spaces with Events of Equal Probability 802.2 Main Principles of Counting 892.3 Permutations 962.4 Combinations 1052.5 The Binomial Theorem 1232.6 Basic Concepts and Formulas 1322.7 Computational Exercises 1332.8 Self-Assessment Exercises 1392.8.1 True–False Questions 1392.8.2 Multiple Choice Questions 1402.9 Review Problems 1432.10 Applications 1502.10.1 Estimation of Population Size: Capture–Recapture Method 150Key Terms 1523 Conditional Probability – Independent Events 1533.1 Conditional Probability 1543.2 The Multiplicative Law of Probability 1663.3 The Law of Total Probability 1743.4 Bayes’ Formula 1833.5 Independent Events 1893.6 Basic Concepts and Formulas 2063.7 Computational Exercises 2073.8 Self-assessment Exercises 2103.8.1 True–False Questions 2103.8.2 Multiple Choice Questions 2113.9 Review Problems 2143.10 Applications 2203.10.1 Diagnostic and Screening Tests 220Key Terms 2234 Discrete Random Variables and Distributions 2254.1 Random Variables 2264.2 Distribution Functions 2324.3 Discrete Random Variables 2474.4 Expectation of a Discrete Random Variable 2614.5 Variance of a Discrete Random Variable 2814.6 Some Results for Expectation and Variance 2934.7 Basic Concepts and Formulas 3024.8 Computational Exercises 3034.9 Self-Assessment Exercises 3094.9.1 True–False Questions 3094.9.2 Multiple Choice Questions 3104.10 Review Problems 3134.11 Applications 3174.11.1 Decision Making Under Uncertainty 317Key Terms 3205 Some Important Discrete Distributions 3215.1 Bernoulli Trials and Binomial Distribution 3225.2 Geometric and Negative Binomial Distributions 3375.3 The Hypergeometric Distribution 3585.4 The Poisson Distribution 3715.5 The Poisson Process 3855.6 Basic Concepts and Formulas 3945.7 Computational Exercises 3955.8 Self-Assessment Exercises 3995.8.1 True–False Questions 3995.8.2 Multiple Choice Questions 4015.9 Review Problems 4035.10 Applications 4115.10.1 Overbooking 411Key Terms 4146 Continuous Random Variables 4156.1 Density Functions 4166.2 Distribution for a Function of a Random Variable 4316.3 Expectation and Variance 4426.4 Additional Useful Results for the Expectation 4516.5 Mixed Distributions 4596.6 Basic Concepts and Formulas 4686.7 Computational Exercises 4696.8 Self-Assessment Exercises 4746.8.1 True–False Questions 4746.8.2 Multiple Choice Questions 4766.9 Review Problems 4796.10 Applications 4866.10.1 Profit Maximization 486Key Terms 4907 Some Important Continuous Distributions 4917.1 The Uniform Distribution 4927.2 The Normal Distribution 5017.3 The Exponential Distribution 5317.4 Other Continuous Distributions 5427.4.1 The Gamma Distribution 5437.4.2 The Beta Distribution 5487.5 Basic Concepts and Formulas 5557.6 Computational Exercises 5577.7 Self-Assessment Exercises 5617.7.1 True–False Questions 5617.7.2 Multiple Choice Questions 5627.8 Review Problems 5657.9 Applications 5737.9.1 Transforming Data: The Lognormal Distribution 573Key Terms 578Appendix A Sums and Products 579Appendix B Distribution Function of the Standard Normal Distribution 593Appendix C Simulation 595Appendix D Discrete and Continuous Distributions 599Bibliography 603Index 605