Details

Introduction to Probability


Introduction to Probability

Multivariate Models and Applications
Wiley Series in Probability and Statistics 1. Aufl.

von: Narayanaswamy Balakrishnan, Markos V. Koutras, Konstadinos G. Politis

111,20 €

Verlag: Wiley
Format: PDF
Veröffentl.: 24.11.2021
ISBN/EAN: 9781118548622
Sprache: englisch
Anzahl Seiten: 544

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Beschreibungen

<b>INTRODUCTION TO PROBABILITY</b> <p><b>Discover practical models and real-world applications of multivariate models useful in engineering, business, and related disciplines</b> <p>In <i>Introduction to Probability: Multivariate Models and Applications, </i>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. <p>This textbook is intended as the sequel to<i> Introduction to Probability: Models and Applications.</i> 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. <p>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: <ul><li>Includes classroom-tested problems and solutions to probability exercises </li> <li>Highlights real-world exercises designed to make clear the concepts presented </li> <li>Uses Mathematica software to illustrate the text’s computer exercises</li> <li>Features applications representing worldwide situations and processes </li> <li>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</li></ul> <p>Perfect for students majoring in statistics, engineering, business, psychology, operations research and mathematics taking a second course in probability, <i>Introduction to Probability: Multivariate Models and Applications</i> is also an indispensable resource for anyone who is required to use multivariate distributions to model the uncertainty associated with random phenomena.
<p>Preface xi</p> <p>Acknowledgments xv</p> <p><b>1 Two-Dimensional Discrete Random Variables and Distributions 1</b></p> <p>1.1 Introduction 2</p> <p>1.2 Joint Probability Function 2</p> <p>1.3 Marginal Distributions 15</p> <p>1.4 Expectation of a Function 24</p> <p>1.5 Conditional Distributions and Expectations 32</p> <p>1.6 Basic Concepts and Formulas 41</p> <p>1.7 Computational Exercises 42</p> <p>1.8 Self-assessment Exercises 46</p> <p>1.8.1 True–False Questions 46</p> <p>1.8.2 Multiple Choice Questions 47</p> <p>1.9 Review Problems 50</p> <p>1.10 Applications 54</p> <p>1.10.1 Mixture Distributions and Reinsurance 54</p> <p>Key Terms 57</p> <p><b>2 Two-Dimensional Continuous Random Variables and Distributions 59</b></p> <p>2.1 Introduction 60</p> <p>2.2 Joint Density Function 60</p> <p>2.3 Marginal Distributions 73</p> <p>2.4 Expectation of a Function 79</p> <p>2.5 Conditional Distributions and Expectations 82</p> <p>2.6 Geometric Probability 91</p> <p>2.7 Basic Concepts and Formulas 98</p> <p>2.8 Computational Exercises 100</p> <p>2.9 Self-assessment Exercises 107</p> <p>2.9.1 True–False Questions 107</p> <p>2.9.2 Multiple Choice Questions 109</p> <p>2.10 Review Problems 111</p> <p>2.11 Applications 114</p> <p>2.11.1 Modeling Proportions 114</p> <p>Key Terms 119</p> <p><b>3 Independence and Multivariate Distributions 121</b></p> <p>3.1 Introduction 122</p> <p>3.2 Independence 122</p> <p>3.3 Properties of Independent Random Variables 137</p> <p>3.4 Multivariate Joint Distributions 142</p> <p>3.5 Independence of More Than Two Variables 156</p> <p>3.6 Distribution of an Ordered Sample 165</p> <p>3.7 Basic Concepts and Formulas 176</p> <p>3.8 Computational Exercises 178</p> <p>3.9 Self-assessment Exercises 185</p> <p>3.9.1 True–False Questions 185</p> <p>3.9.2 Multiple Choice Questions 186</p> <p>3.10 Review Problems 189</p> <p>3.11 Applications 194</p> <p>3.11.1 Acceptance Sampling 194</p> <p>Key Terms 200</p> <p><b>4 Transformations of Variables 201</b></p> <p>4.1 Introduction 202</p> <p>4.2 Joint Distribution for Functions of Variables 202</p> <p>4.3 Distributions of sum, difference, product and quotient 210</p> <p>4.4 <i>𝜒</i><sup>2</sup>, <i>t </i>and <i>F </i>Distributions 223</p> <p>4.5 Basic Concepts and Formulas 236</p> <p>4.6 Computational Exercises 237</p> <p>4.7 Self-assessment Exercises 242</p> <p>4.7.1 True–False Questions 242</p> <p>4.7.2 Multiple Choice Questions 243</p> <p>4.8 Review Problems 246</p> <p>4.9 Applications 250</p> <p>4.9.1 Random Number Generators Coverage – Planning Under Random Event Occurrences 250</p> <p>Key Terms 255</p> <p><b>5 Covariance and Correlation 257</b></p> <p>5.1 Introduction 258</p> <p>5.2 Covariance 258</p> <p>5.3 Correlation Coefficient 272</p> <p>5.4 Conditional Expectation and Variance 281</p> <p>5.5 Regression Curves 293</p> <p>5.6 Basic Concepts and Formulas 307</p> <p>5.7 Computational Exercises 308</p> <p>5.8 Self-assessment Exercises 314</p> <p>5.8.1 True–False Questions 314</p> <p>5.8.2 Multiple Choice Questions 316</p> <p>5.9 Review Problems 320</p> <p>5.10 Applications 326</p> <p>5.10.1 Portfolio Optimization Theory 326</p> <p>Key Terms 330</p> <p><b>6 Important Multivariate Distributions 331</b></p> <p>6.1 Introduction 332</p> <p>6.2 Multinomial Distribution 332</p> <p>6.3 Multivariate Hypergeometric Distribution 344</p> <p>6.4 Bivariate Normal Distribution 358</p> <p>6.5 Basic Concepts and Formulas 371</p> <p>6.6 Computational Exercises 373</p> <p>6.7 Self-Assessment Exercises 378</p> <p>6.7.1 True–False Questions 378</p> <p>6.7.2 Multiple Choice Questions 380</p> <p>6.8 Review Problems 383</p> <p>6.9 Applications 387</p> <p>6.9.1 The Effect of Dependence on the Distribution of the Sum 387</p> <p>Key Terms 390</p> <p><b>7 Generating Functions 391</b></p> <p>7.1 Introduction 392</p> <p>7.2 Moment Generating Function 392</p> <p>7.3 Moment Generating Functions of Some Important Distributions 401</p> <p>7.3.1 Binomial Distribution 401</p> <p>7.3.2 Negative Binomial Distribution 402</p> <p>7.3.3 Poisson Distribution 403</p> <p>7.3.4 Uniform Distribution 403</p> <p>7.3.5 Normal Distribution 403</p> <p>7.3.6 Gamma Distribution 404</p> <p>7.4 Moment Generating Functions for Sum of Variables 407</p> <p>7.5 Probability Generating Function 416</p> <p>7.6 Characteristic Function 428</p> <p>7.7 Generating Functions for Multivariate Case 433</p> <p>7.8 Basic Concepts and Formulas 441</p> <p>7.9 Computational Exercises 443</p> <p>7.10 Self-assessment Exercises 446</p> <p>7.10.1 True–False Questions 446</p> <p>7.10.2 Multiple Choice Questions 448</p> <p>7.11 Review Problems 452</p> <p>7.12 Applications 460</p> <p>7.12.1 Random Walks 460</p> <p>Key Terms 465</p> <p><b>8 Limit Theorems 467</b></p> <p>8.1 Introduction 468</p> <p>8.2 Laws of Large Numbers 468</p> <p>8.3 Central Limit Theorem 476</p> <p>8.4 Basic Concepts and Formulas 492</p> <p>8.5 Computational Exercises 493</p> <p>8.6 Self-assessment Exercises 497</p> <p>8.6.1 True–False Questions 497</p> <p>8.6.2 Multiple Choice Questions 498</p> <p>8.7 Review Problems 501</p> <p>8.8 Applications 504</p> <p>8.8.1 Use of the CLT for Capacity Planning 504</p> <p>Key Terms 507</p> <p>Appendix A Tail Probability Under Standard Normal Distribution 509</p> <p>Appendix B Critical Values Under Chi-Square Distribution 511</p> <p>Appendix C Student’s <i>t</i>-Distribution 515</p> <p>Appendix D <i>F</i>-Distribution: 5% (Lightface Type) and 1% (Boldface Type) Points for the <i>F</i>-Distribution 517</p> <p>Appendix E Generating Functions 521</p> <p>Bibliography 525</p> <p>Index 527</p>
<p><b>N. Balakrishnan, PhD,</b> 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 <i>Encyclopedia of Statistical Sciences, Second Edition.</i> <p><b>Markos V. Koutras, PhD,</b> 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. <p><b>Konstadinos G. Politis, PhD,</b> 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.
<p><b>Discover practical models and real-world applications of multivariate models useful in engineering, business, and related disciplines</b> <p>In <i>Introduction to Probability: Multivariate Models and Applications, </i>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. <p>This textbook is intended as the sequel to<i> Introduction to Probability: Models and Applications.</i> 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. <p>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: <ul><li>Includes classroom-tested problems and solutions to probability exercises</li> <li>Highlights real-world exercises designed to make clear the concepts presented</li> <li>Uses Mathematica software to illustrate the text’s computer exercises</li> <li>Features applications representing worldwide situations and processes</li> <li>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</li></ul> <p>Perfect for students majoring in statistics, engineering, business, psychology, operations research and mathematics taking a second course in probability, <i>Introduction to Probability: Multivariate Models and Applications</i> is also an indispensable resource for anyone who is required to use multivariate distributions to model the uncertainty associated with random phenomena.

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