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E-raamat: Introduction to Probability: Multivariate Models and Applications

(McMaster University, Hamilton, Canada), (University of Piraeus, Greece), (University of Piraeus, Greece)
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INTRODUCTION TO PROBABILITY 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 texts 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.
Preface xi
Acknowledgments xv
1 Two-Dimensional Discrete Random Variables and Distributions
1(58)
1.1 Introduction
2(1)
1.2 Joint Probability Function
2(13)
1.3 Marginal Distributions
15(9)
1.4 Expectation of a Function
24(8)
1.5 Conditional Distributions and Expectations
32(9)
1.6 Basic Concepts and Formulas
41(1)
1.7 Computational Exercises
42(4)
1.8 Self-assessment Exercises
46(4)
1.8.1 True-False Questions
46(1)
1.8.2 Multiple Choice Questions
47(3)
1.9 Review Problems
50(4)
1.10 Applications
54(5)
1.10.1 Mixture Distributions and Reinsurance
54(3)
Key Terms
57(2)
2 Two-Dimensional Continuous Random Variables and Distributions
59(62)
2.1 Introduction
60(1)
2.2 Joint Density Function
60(13)
2.3 Marginal Distributions
73(6)
2.4 Expectation of a Function
79(3)
2.5 Conditional Distributions and Expectations
82(9)
2.6 Geometric Probability
91(7)
2.7 Basic Concepts and Formulas
98(2)
2.8 Computational Exercises
100(7)
2.9 Self-assessment Exercises
107(4)
2.9.1 True-False Questions
107(2)
2.9.2 Multiple Choice Questions
109(2)
2.10 Review Problems
111(3)
2.11 Applications
114(7)
2.11.1 Modeling Proportions
114(5)
Key Terms
119(2)
3 Independence and Multivariate Distributions
121(80)
3.1 Introduction
122(1)
3.2 Independence
122(15)
3.3 Properties of Independent Random Variables
137(5)
3.4 Multivariate Joint Distributions
142(14)
3.5 Independence of More Than Two Variables
156(9)
3.6 Distribution of an Ordered Sample
165(11)
3.7 Basic Concepts and Formulas
176(2)
3.8 Computational Exercises
178(7)
3.9 Self-assessment Exercises
185(4)
3.9.1 True-False Questions
185(1)
3.9.2 Multiple Choice Questions
186(3)
3.10 Review Problems
189(5)
3.11 Applications
194(7)
3.11.1 Acceptance Sampling
194(6)
Key Terms
200(1)
4 Transformations of Variables
201(56)
4.1 Introduction
202(1)
4.2 Joint Distribution for Functions of Variables
202(8)
4.3 Distributions of sum, difference, product and quotient
210(13)
4.4 Χ2, t and F Distributions
223(13)
4.5 Basic Concepts and Formulas
236(1)
4.6 Computational Exercises
237(5)
4.7 Self-assessment Exercises
242(4)
4.7.1 True-False Questions
242(1)
4.7.2 Multiple Choice Questions
243(3)
4.8 Review Problems
246(4)
4.9 Applications
250(7)
4.9.1 Random Number Generators Coverage - Planning Under Random Event Occurrences
250(5)
Key Terms
255(2)
5 Covariance and Correlation
257(74)
5.1 Introduction
258(1)
5.2 Covariance
258(14)
5.3 Correlation Coefficient
272(9)
5.4 Conditional Expectation and Variance
281(12)
5.5 Regression Curves
293(14)
5.6 Basic Concepts and Formulas
307(1)
5.7 Computational Exercises
308(6)
5.8 Self-assessment Exercises
314(6)
5.8.1 True-False Questions
314(2)
5.8.2 Multiple Choice Questions
316(4)
5.9 Review Problems
320(6)
5.10 Applications
326(5)
5.10.1 Portfolio Optimization Theory
326(4)
Key Terms
330(1)
6 Important Multivariate Distributions
331(60)
6.1 Introduction
332(1)
6.2 Multinomial Distribution
332(12)
6.3 Multivariate Hypergeometric Distribution
344(14)
6.4 Bivariate Normal Distribution
358(13)
6.5 Basic Concepts and Formulas
371(2)
6.6 Computational Exercises
373(5)
6.7 Self-Assessment Exercises
378(5)
6.7.1 True-False Questions
378(2)
6.7.2 Multiple Choice Questions
380(3)
6.8 Review Problems
383(4)
6.9 Applications
387(4)
6.9.1 The Effect of Dependence on the Distribution of the Sum
387(3)
Key Terms
390(1)
7 Generating Functions
391(76)
7.1 Introduction
392(1)
7.2 Moment Generating Function
392(9)
7.3 Moment Generating Functions of Some Important Distributions
401(6)
7.3.1 Binomial Distribution
401(1)
7.3.2 Negative Binomial Distribution
402(1)
7.3.3 Poisson Distribution
403(1)
7.3.4 Uniform Distribution
403(1)
7.3.5 Normal Distribution
403(1)
7.3.6 Gamma Distribution
404(3)
7.4 Moment Generating Functions for Sum of Variables
407(9)
7.5 Probability Generating Function
416(12)
7.6 Characteristic Function
428(5)
7.7 Generating Functions for Multivariate Case
433(8)
7.8 Basic Concepts and Formulas
441(2)
7.9 Computational Exercises
443(3)
7.10 Self-assessment Exercises
446(6)
7.10.1 True-False Questions
446(2)
7.10.2 Multiple Choice Questions
448(4)
7.11 Review Problems
452(8)
7.12 Applications
460(7)
7.12.1 Random Walks
460(5)
Key Terms
465(2)
8 Limit Theorems
467(42)
8.1 Introduction
468(1)
8.2 Laws of Large Numbers
468(8)
8.3 Central Limit Theorem
476(16)
8.4 Basic Concepts and Formulas
492(1)
8.5 Computational Exercises
493(4)
8.6 Self-assessment Exercises
497(4)
8.6.1 True-False Questions
497(1)
8.6.2 Multiple Choice Questions
498(3)
8.7 Review Problems
501(3)
8.8 Applications
504(5)
8.8.1 Use of the CLT for Capacity Planning
504(3)
Key Terms
507(2)
Appendix A Tail Probability Under Standard Normal Distribution 509(2)
Appendix B Critical Values Under Chi-Square Distribution 511(4)
Appendix C Student's r-Distribution 515(2)
Appendix D F-Distribution: 5% (Lightface Type) and 1% (Boldface Type) Points for the F-Distribution 517(4)
Appendix E Generating Functions 521(4)
Bibliography 525(2)
Index 527
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.
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