概率论与数理统计Morris H. Degroot

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概率论与数理统计Morris H. Degroot
To Jenny and Jeremy Preface This book contains enough material for a one-year course in probability and statistics. The mathematical requirements for the course are a knowledge of the elements of calculus and a familiarity with the concepts and elementary proper ties of vectors and matrices no previous knowledge of probability or statistics is assume d The book has been written with both the student and the teacher in mind Special care has been taken to make sure that the text can be read and understood with few obscure passages or other stumbling blockS. Theorems and proofs are presented where appropriate, and illustrative examples are given at almost every step of the way. More than 1100 exercises are included in the book. Some of these exercises provide numerical applications of results presented in the text, and others are intended to stimulate further thought about these results. A new feature of this second edition is the inclusion of approximately 20 or 25 exercises at the end of each chapter that supplement the exercises given at the end of most of the individual sections of the book The first five chapters are devoted to probability and can serve as the text for a one-semester course on that topic. The elementary concepts of probability are illustrated by such famous examples as the birthday problem, the tennis tourn ment problem, the matching problem, the collector's problem, and the game of craps. Standard material on random variables and probability distributions is highlighted by discussions of statistical swindles, the use of a table of random digits, the elementary notions of life testing, a comparison of the relative advantages of the mean and the median as predictors, the importance of the central limit theorem, and the correction for continuity. Also included as special features of these chapters are sections on Markov chains, the Gamblers ruin Preface problem, choosing the best, utility and preferences among gambles, and the Borel-Kolmogorov paradox. These topics are treated in a completely elemcntary fashion, but they can be omitted without loss of continuity if time is limited. Sections of the book that can be so omitted are indicated, in the traditional way, by asterisks in the Contents and in the text CO. The last five chapters of the book are devoted to statistical inference.The crage here is modern in outlook. Both classical and Bayesian statistical methods are developed in an integrated presentation. No single school of thought is treated in a dogmatic fashion. My goal is to equip the student with the theor and methodology that have proved to be useful in the past and promise to bc useful in the future tesx These chapters contain a comprehensive but elementary survey of estimation, testing hypotheses, nonparamctric methods, multiple regression, and the analysis of variance. The strengths and weaknesses and the advantages and disadvantages of such basic concepts as maximum likelihood estimation, Bayesian decision procedures, unbiased estimation, confidence intervals, and levels of significance are discussed from a contemporary viewpoint. Special features of these chapters include discussions of prior and posterior distributions, sufficient statistics, Fisher information, the delta method, the Bayesian analysis of samples from a normal distribution. unbiased tests, multidecision problems, tests of goodness-of-fit, contingency tables, Simpson's paradox, inferences about the median and other quantiles, robust estimation and trimmed means, confidence bands for a regres- sion line, and the regression fallacy. If time does not permit complete coverage of the contents of these chapters, any of the following sections can be omitted without loss of continuity: 7.6, 7.8, 8.3, 9.6, 9.7, 9.8, 99, and 9.10 In summary, the main changes in this second edition are new sections or subsections on statistical swindles, choosing the best, the Borel-Kolmogorov paradox, the correction for continuity, the delta. method. unbia ased tests, Simpson’s paradox, confidence bands for a regression line, and the regression fallacy, as well as a new section of supplementary exercises at the end of each chapter. The material introducing random variables and their distributions has been thor- oughly revised, and minor changes, additions, and deletions have been made throughout the text Although a computer can be a valuable adjunct in a course in probability and statistics such as this one none of the exercises in this book requires access to a computer or a knowledge of programming. For this reason, the use of this book is not tied to a computer in any way. Instructors are urged, however, to use computers in the course as much as is feasible. A small calculator is a helpful aid for solving some of the numerical exercises in the second half of the book One further point about the style in which the book is written should be emphasized. The pronoun "he"is used throughout the book in reference to a person who is confronted with a statistical problem. This usage certainly does not mean that only males calculate probabilities and make decisions, or that only Preface males can be statisticians. The word "he"is used quite literally as defined in Webster's Third. New International Dictionary to mean "that one whose sex is unknown or immaterial. "The field of statistics should certainly be as accessible to women as it is to men. It should certainly be as accessible to members of minority groups as it is to thc majority. It is my sincere hope that this book will help create among all groups an awareness and appreciation of probability and statistics as an interesting, lively, and important branch of science I am indebted to the readers, instructors, and colleagues whose comments have strengthened this edition. Marion Reynolds, r, of Virginia Polytechnic Institute and James Stapleton of Michigan State University reviewed the manuscript for the publisher and made many valuable suggestions. I am grateful to the Literary Executor of the late Sir Ronald A. Fisher, F.R.S., to Dr. Frank Yates, F.R.S., and the Longman group Ltd,, London, for permission to adapt Table iii of their book Statistical Tables for Biological, Agricultural and medical Research (6th edition, 1974) The field of statistics has grown and changed since I wrote a Preface for the first edition of this book in November, 1974, and so have I. The influence on my life and work of those who made that first edition possible remains vivid and undiminished; but with growth and change have come new influences as well both personal and professional. The love, warmth, and support of my family and friends old and new. have sustained and stimulated me, and enabled me to write a book that i believe reflects contemporary probability and statistics Pittsburgh, Pennsylvania M.H. D October 1985 Contents Introduction to Probability The history of Probability 1 Interpretations of Probability 2 Experiments and Events 5 Set Theory 7 The Definition of Probability 13 Finite Sample Spaces 18 1.7 Counting methods 20 Combinatorial Methods 26 1.9 Multinomial Coefficients 32 1.10 The Probability of a Union of Events 36 1.11 Independent Events 43 1.12 Statistical Swindles 52 1.13 Supplementary Exercises 54 2 Conditional Probability 2.1 Thc Dcfinition of Conditional Probability 57 2.2 Bayes'Theorem 64 x 2.3 Markov Chains 72 24 The Gambler's Ruin Problem 82 viii Contents 25 Choosing the Best 87 26 Supplementary Exercises 94 3 Random variables and distributions Random variables and Discrete Distributions 97 3.2 Continuous Distributions 102 3.3 The Distribution Function 108 3.4 Bivariate distributions 115 3.5 Marginal Distributions 125 Conditional distributions 134 3.7 Multivariate Distributions 142 3.8 Functions of a Random variable 150 39 Functions of Two or more random variables 158 3. 10 The Borel-Kolmogorov Paradox 171 3. 11 Supplementary Exercises 174 4 Expectation 4.1 The Expectation of a Random Variable 179 4.2 Properties of Expectations 187 4.3 Variance 194 4.4 Moments 199 4.5 The mean and the Median 206 4.6 Covariance and Correlation 213 4.7 Conditional Expectation 219 4.8 The Sample Mean 226 4.9 Utility 233 4.10 Supplementary Exercises 239 5 Special Distributions 5.1 Introduction 243 5.2 The Bernoulli and Binomial Distributions 243 5.3 The Hypergeometric Distribution 247 The poisson distribution 252 5.5 The Negative Binomial Distribution 258 5.6 The normal Distribution 263 The Central Limit Theorem 274 Contents 5.8 The Correction for Continuity 283 59 The Gamma Distribution 286 5.10 The Beta Distribution 294 5.11 The Multinomial Distribution 297 5. 12 The Bivariate Normal Distribution 300 5. 13 Supplementary Exercises 307 Estimation 6.1 Statistical Inference 311 6.2 Prior and Posterior distributions 313 6.3 Conjugate Prior Distributions 321 64 Bayes Estimators 330 6.5 Maximum likelihood Estimators 338 6.6 Properties of Maximum Likelihood Estimators 348 Sufficient statistics 356 68 Jointly Sufficient Statistics 3 64 Improving an Estimator 371 6.10 Supplementary Exercises 377 SamplIng distributions of Estimators 7.1 The Sampling Distribution of a Statistic 381 72 The Chi-Square Distribution 383 73 Joint Distribution of the Sample mean and Sample variance 386 7,4 The t Distribution 393 7.5 Confidence Intervals 398 7,6 Bayesian Analysis of Samples from a Normal Distribution 402 7.7 Unbiased Estimators 411 78 Fisher Information 420 7.9 Supplementary Exercises 433 8 Testing Hypotheses 8.1 Problems of Testing Hypotheses 437 8.2 Testing Simple Hypotheses 442 4f 8. 3 Multidecision problems 456 Contents 8.4 Uniformly most Powerful Tests 466 8.5 Selecting a 'Test Procedure 477 86 The t Test 485 8.7 Discussion of the Methodology of Testing Hypotheses 494 8.8 The f Distribution 499 89 Comparing the means of Two Normal Distributions 506 8.10 Supplementary Exercises 512 9 Categorical Data and Nonparametric Methods 91 Tests of oodness-of-Fit 519 9,2 Goodness-of-Fit for Composite Hypotheses 526 9.3 Contingency Tables 534 94 Tests of Homogeneity 540 9.5 Simpsons Paradox 548 *96 Kolmogorov-Smirnov Tests 552 *97 Inferences about the Median and Other Quantiles 561 *9.8 Robust Estimation 564 Paired Observations 571 910 Ranks for Two Samples 580 9.11 Supplementary Exercises 586 10 Linear Statistical Models 10.1 The Method of Least Squares 593 10.2 Regression 604 10.3 Tests of Hypotheses and Confidence Intervals in Simple Linear Regression 612 104 The Regression Fallacy 628 10.5 Multiple regi 631 10.6 Analysis of Variance 644 10.7 The Two-Way Layout 652 10. 8 The Two-Way layout with Ref cations 662 109 Supplementary Exercises 6/7 li References 679 Tables Binomial Probabilities 682 Contents Random digits 685 Poisson Probabilities 688 The Standard Normal Distribution Function 689 The x Distribution 690 The t Distribution 692 0.95 Quantile of the F Distribution 694 0.975 Quantile of the F Distribution 695 Answers to Even-Numbered Exercises 697 Index 717

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