Hogg Craig Introduction to Mathematical Statistics

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Introduction to

Mathematical Statistics

Eighth Edition

Robert V. Hogg

University of Iowa

Joseph W. McKean

Western Michigan University

Allen T. Craig

Late Professor of Statistics University of Iowa

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Library of Congress Cataloging-in-Publications Data Names: Hogg, Robert V., author. | McKean, Joseph W., 1944- author. | Craig,

Allen T. (Allen Thornton), 1905- author. Title: Introduction to mathematical statistics / Robert V. Hogg, Late Professor of Statistics,

University of Iowa, Joseph W. McKean, Western Michigan University, Allen T. Craig, Late Professor of Statistics, University of Iowa. Description: Eighth edition. | Boston : Pearson, [2019] | Includes bibliographical references and index. Identifiers: LCCN 2017033015| ISBN 9780134686998 | ISBN 0134686993 Subjects: LCSH: Mathematical statistics. Classification: LCC QA276 .H59 2019 | DDC 519.5?dc23 LC record available at

ISBN 13: 978-0-13-468699-8

ISBN 10: 0-13-468699-3

Dedicated to my wife Marge and to the memory of Bob Hogg

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Contents

Preface

xi

1 Probability and Distributions

1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Review of Set Theory . . . . . . . . . . . . . . . . . . . . . . 4

1.2.2 Set Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 The Probability Set Function . . . . . . . . . . . . . . . . . . . . . . 12

1.3.1 Counting Rules . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.3.2 Additional Properties of Probability . . . . . . . . . . . . . . 18

1.4 Conditional Probability and Independence . . . . . . . . . . . . . . . 23

1.4.1 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.4.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

1.5 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

1.6 Discrete Random Variables . . . . . . . . . . . . . . . . . . . . . . . 45

1.6.1 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . 47

1.7 Continuous Random Variables . . . . . . . . . . . . . . . . . . . . . 49

1.7.1 Quantiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

1.7.2 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . 53

1.7.3 Mixtures of Discrete and Continuous Type Distributions . . . 56

1.8 Expectation of a Random Variable . . . . . . . . . . . . . . . . . . . 60

1.8.1 R Computation for an Estimation of the Expected Gain . . . 65

1.9 Some Special Expectations . . . . . . . . . . . . . . . . . . . . . . . 68

1.10 Important Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 78

2 Multivariate Distributions

85

2.1 Distributions of Two Random Variables . . . . . . . . . . . . . . . . 85

2.1.1 Marginal Distributions . . . . . . . . . . . . . . . . . . . . . . 89

2.1.2 Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

2.2 Transformations: Bivariate Random Variables . . . . . . . . . . . . . 100

2.3 Conditional Distributions and Expectations . . . . . . . . . . . . . . 109

2.4 Independent Random Variables . . . . . . . . . . . . . . . . . . . . . 117

2.5 The Correlation Coefficient . . . . . . . . . . . . . . . . . . . . . . . 125

2.6 Extension to Several Random Variables . . . . . . . . . . . . . . . . 134

v

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Contents

2.6.1 Multivariate Variance-Covariance Matrix . . . . . . . . . . . 140 2.7 Transformations for Several Random Variables . . . . . . . . . . . . 143 2.8 Linear Combinations of Random Variables . . . . . . . . . . . . . . . 151

3 Some Special Distributions

155

3.1 The Binomial and Related Distributions . . . . . . . . . . . . . . . . 155

3.1.1 Negative Binomial and Geometric Distributions . . . . . . . . 159

3.1.2 Multinomial Distribution . . . . . . . . . . . . . . . . . . . . 160

3.1.3 Hypergeometric Distribution . . . . . . . . . . . . . . . . . . 162

3.2 The Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . . 167 3.3 The , 2, and Distributions . . . . . . . . . . . . . . . . . . . . . 173

3.3.1 The 2-Distribution . . . . . . . . . . . . . . . . . . . . . . . 178

3.3.2 The -Distribution . . . . . . . . . . . . . . . . . . . . . . . . 180

3.4 The Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 186 3.4.1 Contaminated Normals . . . . . . . . . . . . . . . . . . . . . 193

3.5 The Multivariate Normal Distribution . . . . . . . . . . . . . . . . . 198

3.5.1 Bivariate Normal Distribution . . . . . . . . . . . . . . . . . . 198 3.5.2 Multivariate Normal Distribution, General Case . . . . . . . 199 3.5.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

3.6 t- and F -Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 210

3.6.1 The t-distribution . . . . . . . . . . . . . . . . . . . . . . . . 210

3.6.2 The F -distribution . . . . . . . . . . . . . . . . . . . . . . . . 212

3.6.3 Student's Theorem . . . . . . . . . . . . . . . . . . . . . . . . 214 3.7 Mixture Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 218

4 Some Elementary Statistical Inferences

225

4.1 Sampling and Statistics . . . . . . . . . . . . . . . . . . . . . . . . . 225

4.1.1 Point Estimators . . . . . . . . . . . . . . . . . . . . . . . . . 226

4.1.2 Histogram Estimates of pmfs and pdfs . . . . . . . . . . . . . 230

4.2 Confidence Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

4.2.1 Confidence Intervals for Difference in Means . . . . . . . . . . 241

4.2.2 Confidence Interval for Difference in Proportions . . . . . . . 243 4.3 Confidence Intervals for Parameters of Discrete Distributions . . . . 248

4.4 Order Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

4.4.1 Quantiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

4.4.2 Confidence Intervals for Quantiles . . . . . . . . . . . . . . . 261

4.5 Introduction to Hypothesis Testing . . . . . . . . . . . . . . . . . . . 267

4.6 Additional Comments About Statistical Tests . . . . . . . . . . . . . 275

4.6.1 Observed Significance Level, p-value . . . . . . . . . . . . . . 279

4.7 Chi-Square Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

4.8 The Method of Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . 292

4.8.1 Accept?Reject Generation Algorithm . . . . . . . . . . . . . . 298

4.9 Bootstrap Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . 303

4.9.1 Percentile Bootstrap Confidence Intervals . . . . . . . . . . . 303

4.9.2 Bootstrap Testing Procedures . . . . . . . . . . . . . . . . . . 308 4.10 Tolerance Limits for Distributions . . . . . . . . . . . . . . . . . . . 315

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