Discover practical models and real-world applications of multivariate models useful in engineering, business, and related disciplines
In Introduction to Probability: Multivariate Models and Applications, a team of distinguished researchers delivers a comprehensive exploration of the concepts, methods, and results in multivariate distributions and models. Intended for use in a second course in probability, the material is largely self-contained, with some knowledge of basic probability theory and univariate distributions as the only prerequisite.
This textbook is intended as the sequel to Introduction to Probability: Models and Applications. Each chapter begins with a brief historical account of some of the pioneers in probability who made significant contributions to the field. It goes on to describe and explain a critical concept or method in multivariate models and closes with two collections of exercises designed to test basic and advanced understanding of the theory.
A wide range of topics are covered, including joint distributions for two or more random variables, independence of two or more variables, transformations of variables, covariance and correlation, a presentation of the most important multivariate distributions, generating functions and limit theorems. 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
Perfect for students majoring in statistics, engineering, business, psychology, operations research and mathematics taking a second course in probability, Introduction to Probability: Multivariate Models and Applications is also an indispensable resource for anyone who is required to use multivariate distributions to model the uncertainty associated with random phenomena.
Table of Contents
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 𝜒2, 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
