SIXTH EDITION

John E. Freund's Mathematical Statistics

IRWIN MILLER

MARYLEES MILLER

Prentice Hall International, Inc.

Contents

PREFACE xi

1 INTRODUCTION 1

1.1 Introduction 1 1.2 Combinatorial Methods 2

1.3 Binomial Coefficients 12

2 PROBABILITY 25

2.1 Introduction 25 2.2 Sample Spaces 26 2.3 Events 28 2.4 The Probability of an Event 36 2.5 Some Rules of Probability 42 2.6 Conditional Probability 52 2.7 Independent Events 58 2.8 Bayes' Theorem 62

3 PROBABIUTY DISTRIBUTIONS AND PROBABILITY DENSITIES 73

3.1 Introduction 73 3.2 Probability Distributions 77 3.3 Continuous Random Variables 89 3.4 Probability Density Functions 90 3.5 Multivariate Distributions 102 3.6 Marginal Distributions 115 3.7 Conditional Distributions 119

V

vi Contents

4 MATHEMATICAL EXPECTATION 129

4.1 Introduction 129 4.2 The Expected Value of a Random Variable 1 30 4.3 Moments 140 4.4 Chebyshev's Theorem 144 4.5 Moment-Generating Functions 146 4.6 Product Moments 153 4.7 Moments of Linear Combinations of Random

Variables 158 4.8 Conditional Expectations 161

5 SPECIAL PROBABILITY DISTRIBUTIONS 167

5.1 Introduction 167 5.2 The Discrete Uniform Distribution 167 5.3 The Bernoulli Distribution 168 5.4 The Binomial Distribution 169 5.5 The Negative Binomial and Geometrie

Distributions 180 5.6 The Hypergeometric Distribution 182 5.7 The Poisson Distribution 186 5.8 The Multinomial Distribution 198 5.9 The Multivariate Hypergeometric Distribution 200

6 SPECIAL PROBABILITY DENSITIES 203

6.1 Introduction 203 6.2 The Uniform Distribution 203 6.3 The Gamma, Exponential, and Chi-Square

Distributions 204 6.4 The Beta Distribution 210 6.5 The Normal Distribution 216 6.6 The Normal Approximation to the Binomial

Distribution 222 6.7 The Bivariate Normal Distribution 229

7 FUNCTIONS OF RANDOM VARIABLES 236

7.1 Introduction 236 7.2 Distribution Function Technique 237 7.3 Transformation Technique: One Variable 242 7.4 Transformation Technique: Several Variables 249 7.5 Moment-Generating Function Technique 261

Contents vii

8 SAMPLING DISTRIBUTIONS 266

8.1 Introduction 266 8.2 The Distribution of the Mean 268 8.3 The Distribution of the Mean: Finite

Populations 272 8.4 The Chi-Square Distribution 279 8.5 The t Distribution 283 8.6 The F Distribution 286 8.7 Order Statistics 293

9 DECISION THEORY 300

9.1 Introduction 300 9.2 The Theory of Games 302 9.3 Statistical Games 312 9.4 Decision Criteria 315 9.5 The Minimax Criterion 316 9.6 The Bayes Criterion 317

10 ESTIMATION: THEORY 322

10.1 Introduction 322 10.2 Unbiased Estimators 323 10.3 Efficiency 326 10.4 Consistency 335 10.5 Sufficiency 337 10.6 Robustness 341 10.7 The Method of Moments 343 10.8 The Method of Maximum Likelihood 345 10.9 Bayesian Estimation 353

/1 ESTIMATION: APPLICATIONS 360

11.1 Introduction 360 11.2 The Estimation of Means 361 11.3 The Estimation of Differences Between Means 365 11.4 The Estimation of Proportions 372 11.5 The Estimation of Differences Between

Proportions 374 11.6 The Estimation of Variances 378 11.7 The Estimation of the Ratio of Two Variances 379 11.8 Use of Computers 381

Contents

12 HYPOTHESIS TE5TINC: THEORY 384

12.1 Introduction 384 12.2 Testing a Statistical Hypothesis 386 12.3 Losses and Risks 388 12.4 The Neyman-Pearson Lemma 389 12.5 The Power Function of a Test 397 12.6 Likelihood Ratio Tests 400

13 HYPOTHESIS TESTINC: APPLICATIONS 410

13.1 Introduction 410 13.2 Tests Concerning Means 415 1 3.3 Tests Concerning Differences Between Means 418 1 3.4 Tests Concerning Variances 426 13.5 Tests Concerning Proportions 430 1 3.6 Tests Concerning Differences Among k

Proportions 432 13.7 The Analysisofan r X c Table 438 13.8 Goodness of Fit 441 1 3.9 Use of Computers 446

14 REGRESSION AND CORRELATION 449

14.1 Introduction 449 14.2 Linear Regression 453 14.3 The Method of Least Squares 455 14.4 Normal Regression Analysis 464 14.5 Normal Correlation Analysis 473 14.6 Multiple Linear Regression 480 14.7 Multiple Linear Regression (Matrix Notation) 484

15 ANALYSIS OF VARIANCE 496

15.1 Introduction 496 15.2 One-Way Analysis of Variance 496 15.3 Experimental Design 504 15.4 Two-Way Analysis of Variance Without

Interaction 506 15.5 Two-Way Analysis of Variance with Interaction 514 15.6 Multiple Comparisons 522 15.7 Some Further Considerations 525

Contents ix

16 NONPARAMETRIC TESTS 527

16.1 Introduction 527 16.2 The Sign Test 529 16.3 The Signed-Rank Test 531 16.4 Rank-Sum Tests: The U Test 539 16.5 Rank-Sum Tests: The H Test 543 16.6 Tests Based on Runs 548 16.7 The Rank Correlation Coefficient 554

APPENDIX A: SUMS AND PRODUCTS 560

A.1 Rules for Sums and Products 560 A.2 Special Sums 561

APPENDIX B: SPECIAL PROBABILITY DISTRIBUTIONS 564

APPENDIX C: SPECIAL PROBABILITY DENSITIES 566

STATISTICAL TABLES 569

ANSWERS TO ODD-NUMBERED EXERCISES 595

INDEX 614