Probability statistical inference hogg pdf

PROBABILITY AND STATISTICAL INFERENCE Ninth Edition

Robert V. Hogg

Elliot A. Tanis

Dale L. Zimmerman

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Library of Congress Cataloging-in-Publication Data Hogg, Robert V.

Probability and Statistical Inference/ Robert V. Hogg, Elliot A. Tanis, Dale Zimmerman. – 9th ed.

p. cm. ISBN 978-0-321-92327-1

1. Mathematical statistics. I. Hogg, Robert V., II. Tanis, Elliot A. III. Title. QA276.H59 2013 519.5–dc23

2011034906

10 9 8 7 6 5 4 3 2 1 EBM 17 16 15 14 13

www.pearsonhighered.com ISBN-10: 0-321-92327-8 ISBN-13: 978-0-321-92327-1

http://www.pearsonhighered.com

Contents Preface v

Prologue vii

1 Probability 1 1.1 Properties of Probability 1

1.2 Methods of Enumeration 11

1.3 Conditional Probability 20

1.4 Independent Events 29

1.5 Bayes’ Theorem 35

2 Discrete Distributions 41 2.1 Random Variables of the Discrete Type 41

2.2 Mathematical Expectation 49

2.3 Special Mathematical Expectations 56

2.4 The Binomial Distribution 65

2.5 The Negative Binomial Distribution 74

2.6 The Poisson Distribution 79

3 Continuous Distributions 87 3.1 Random Variables of the Continuous

Type 87

3.2 The Exponential, Gamma, and Chi-Square Distributions 95

3.3 The Normal Distribution 105

3.4* Additional Models 114

4 Bivariate Distributions 125 4.1 Bivariate Distributions of the Discrete

Type 125

4.2 The Correlation Coefficient 134

4.3 Conditional Distributions 140

4.4 Bivariate Distributions of the Continuous Type 146

4.5 The Bivariate Normal Distribution 155

5 Distributions of Functions of Random Variables 163

5.1 Functions of One Random Variable 163

5.2 Transformations of Two Random Variables 171

5.3 Several Random Variables 180

5.4 The Moment-Generating Function Technique 187

5.5 Random Functions Associated with Normal Distributions 192

5.6 The Central Limit Theorem 200

5.7 Approximations for Discrete Distributions 206

5.8 Chebyshev’s Inequality and Convergence in Probability 213

5.9 Limiting Moment-Generating Functions 217

6 Point Estimation 225 6.1 Descriptive Statistics 225

6.2 Exploratory Data Analysis 238

6.3 Order Statistics 248

6.4 Maximum Likelihood Estimation 256

6.5 A Simple Regression Problem 267

6.6* Asymptotic Distributions of Maximum Likelihood Estimators 275

6.7 Sufficient Statistics 280

6.8 Bayesian Estimation 288

6.9* More Bayesian Concepts 294

7 Interval Estimation 301 7.1 Confidence Intervals for Means 301

7.2 Confidence Intervals for the Difference of Two Means 308

7.3 Confidence Intervals for Proportions 318

7.4 Sample Size 324 iii

iv Contents

7.5 Distribution-Free Confidence Intervals for Percentiles 331

7.6* More Regression 338

7.7* Resampling Methods 347

8 Tests of Statistical Hypotheses 355

8.1 Tests About One Mean 355

8.2 Tests of the Equality of Two Means 365

8.3 Tests About Proportions 373

8.4 The Wilcoxon Tests 381

8.5 Power of a Statistical Test 392

8.6 Best Critical Regions 399

8.7* Likelihood Ratio Tests 406

9 More Tests 415 9.1 Chi-Square Goodness-of-Fit Tests 415

9.2 Contingency Tables 424

9.3 One-Factor Analysis of Variance 435

9.4 Two-Way Analysis of Variance 445

9.5* General Factorial and 2k Factorial Designs 455

9.6* Tests Concerning Regression and Correlation 462

9.7* Statistical Quality Control 467

Epilogue 479

Appendices

A References 481 B Tables 483 C Answers to Odd-Numbered

Exercises 509

D Review of Selected Mathematical Techniques 521

D.1 Algebra of Sets 521

D.2 Mathematical Tools for the Hypergeometric Distribution 525

D.3 Limits 528

D.4 Infinite Series 529

D.5 Integration 533

D.6 Multivariate Calculus 535

Index 541

Preface

In this Ninth Edition of Probability and Statistical Inference, Bob Hogg and Elliot Tanis are excited to add a third person to their writing team to contribute to the continued success of this text. Dale Zimmerman is the Robert V. Hogg Professor in the Department of Statistics and Actuarial Science at the University of Iowa. Dale has rewritten several parts of the text, making the terminology more consistent and contributing much to a substantial revision. The text is designed for a two-semester course, but it can be adapted for a one-semester course. A good calculus background is needed, but no previous study of probability or statistics is required.

CONTENT AND COURSE PLANNING In this revision, the first five chapters on probability are much the same as in the eighth edition. They include the following topics: probability, conditional probability, independence, Bayes’ theorem, discrete and continuous distributions, certain math- ematical expectations, bivariate distributions along with marginal and conditional distributions, correlation, functions of random variables and their distributions, including the moment-generating function technique, and the central limit theorem. While this strong probability coverage of the course is important for all students, it has been particularly helpful to actuarial students who are studying for Exam P in the Society of Actuaries’ series (or Exam 1 of the Casualty Actuarial Society).

The greatest change to this edition is in the statistical inference coverage, now Chapters 6–9. The first two of these chapters provide an excellent presentation of estimation. Chapter 6 covers point estimation, including descriptive and order statistics, maximum likelihood estimators and their distributions, sufficient statis- tics, and Bayesian estimation. Interval estimation is covered in Chapter 7, including the topics of confidence intervals for means and proportions, distribution-free con- fidence intervals for percentiles, confidence intervals for regression coefficients, and resampling methods (in particular, bootstrapping).

The last two chapters are about tests of statistical hypotheses. Chapter 8 consid- ers terminology and standard tests on means and proportions, the Wilcoxon tests, the power of a test, best critical regions (Neyman/Pearson) and likelihood ratio tests. The topics in Chapter 9 are standard chi-square tests, analysis of variance including general factorial designs, and some procedures associated with regression, correlation, and statistical quality control.

The first semester of the course should contain most of the topics in Chapters 1–5. The second semester includes some topics omitted there and many of those in Chapters 6–9. A more basic course might omit some of the (optional) starred sections, but we believe that the order of topics will give the instructor the flexibility needed in his or her course. The usual nonparametric and Bayesian techniques are placed at appropriate places in the text rather than in separate chapters. We find that many persons like the applications associated with statistical quality control in the last section. Overall, one of the authors, Hogg, believes that the presentation (at a somewhat reduced mathematical level) is much like that given in the earlier editions of Hogg and Craig (see References).

v

vi Preface

The Prologue suggests many fields in which statistical methods can be used. In the Epilogue, the importance of understanding variation is stressed, particularly for its need in continuous quality improvement as described in the usual Six-Sigma pro- grams. At the end of each chapter we give some interesting historical comments, which have proved to be very worthwhile in the past editions.

The answers given in this text for questions that involve the standard distribu- tions were calculated using our probability tables which, of course, are rounded off for printing. If you use a statistical package, your answers may differ slightly from those given.

ANCILLARIES Data sets from this textbook are available on Pearson Education’s Math & Statistics Student Resources website: http://www.pearsonhighered.com/mathstatsresources.

An Instructor’s Solutions Manual containing worked-out solutions to the even- numbered exercises in the text is available for download from Pearson Education’s Instructor Resource Center at www.pearsonhighered.com/irc. Some of the numer- ical exercises were solved with Maple. For additional exercises that involve sim- ulations, a separate manual, Probability & Statistics: Explorations with MAPLE, second edition, by Zaven Karian and Elliot Tanis, is also available for download from Pearson Education’s Instructor Resource Center. Several exercises in that manual also make use of the power of Maple as a computer algebra system.

If you find any errors in this text, please send them to tanis@hope.edu so that they can be corrected in a future printing. These errata will also be posted on http://www.math.hope.edu/tanis/.