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Student's t X t(v) v= 1,2,... Snedecor's F X F(v1, y2) Beta X - BETA(a,b) O <a 0<b *Not tractable. Does not exist. Specia' Continuous Distributions (y1 + v2 2 ) (vii! (y \ Ív2\ '\!'2) \\2) \2 O l<v V v-2 2<v F'(a)F(b) O<x<1 a a+b ab (a+b+ l)(a+b)2 Notation and Parameters Continuous pdf fix) Mean Variance MGF M(t) 2v(v1 +v2-2) v1(v2 2)2(v2 4) 2 y1 = 1,2,. y2 = 1,2,. 2<v2 4<v2 \ VJ PDF compression, OCR, web optimization using a watermarked evaluation copy of CVISION PDFCompressor Cauchy X CAU(O, ,j) 0<0 Pareto X - PAR(O, ,c) 0<0 Specia' Continuous Distributions thr{1 + [(x )/0],} ¡ç 0(1 + X/O)K +1 0<x y 0.5772 (Euler's const.) O 02K Chi-Square X -'- x2(v) X''2 'e'2 V 2v 2/2 U(v/2) v=1,2,... 0<x ß ** I 2t Notation and Parameters Continuous pdf f(x) Mean Variance MGF M(t) WeibuH X WEI(0,ß) xe' or(i +) 02[(t + - 0<0 0<x 0<ß Extreme Value X EV(0, 17) exp {[(x )/O] exp [(x-17)/0]} - yO 0<0 'ç1 (-2)( 1)2 1<K 2< 2O2 6 &T(i + Ot) PDF compression, OCR, web optimization using a watermarked evaluation copy of CVISION PDFCompressor INTRODUCTION TO PROBABILITY AND MATHEMATICAL STATISTICS SECOND EDITION Lee J. Bain University of Missouri - Rol/a Max Engelhardt University of Idaho At Du::bur Thomson Learñing Australia Canada Mexico Singapore Spain United Kingdom United States PDF compression, OCR, web optimization using a watermarked evaluation copy of CVISION PDFCompressor The Duxbuiy Classic Series is a collection of authoritativeworks from respected authors. Reissued as paperbacks, these successful titles are now more affordable. COPYRIGHT © 1992, 1987 by Brooks/Cole Dux bury is an imprint of Brooks/Cole, a division of Thomson Learning The Thomson Learning logo is a trademark used herein under license. For more information about this or any óther Duxbury product, contact: DUXBTJRY 511 Forest Lodge Road Pacific Grove, CA 93950 USA www.duxbury.com 1-800-423-0563 (Thomson Learning Academic Resource Center) All rights reserved. No part of this work may be reproduced, transcribed or used in any form or by any meansgraphic, electronic, or mechanical, including photocopying, recording, taping, Web distribution, or information storage and/or retrieval systemswithout the prior written permission of the publisher. For permission to use material from this work, contact us by Web: www.thomsonrights.com fax: 1-800-730-2215 phone: 1-800-730-2214 Printed in the United States of America 10 9 8 7 6 5 4 3 2 Library of Congress Cataloging-in-Publication Data Bain, Lee J. Introduction to probability and mathematical statistics / Lee J. Bain, Max Engelhardt.-2°' ed. p. cm.(The Duxbuiy advanced series in statistics and decision sciences) Includes bibliographical references and index. ISBN 0-534-92930-3 (hard cover) ISBN 0-534-38020-4 (paperback) 1. Probabilities. 2. Mathematical statistics. I. Engelhardt, Max. II. Title. ifi. Series. QA273.B2546 1991 519.2---dc2O 91-25923 dF PDF compression, OCR, web optimization using a watermarked evaluation copy of CVISION PDFCompressor CONTENTS CHA PTER PROBABILITY i 1.1 Introduction 1 1.2 Notation and terminology 2 1.3 Definition of probability 9 1.4 Some properties of probability 13 1.5 Conditional probability 16 1.6 Counting techniques 31 Summary 42 Exercises 43 CHAPTER RANDOM VARIABLES AND THEIR DISTRIBUTIONS 53 2.1 Introduction 53 2.2 Discrete random variables 56 2.3 Continuous random variables 62 2.4 Some properties of expected values 71 2.5 Moment generating functions 78 Summary 83 Exercises 83 V PDF compression, OCR, web optimization using a watermarked evaluation copy of CVISION PDFCompressor Vi CONTENTS CHAPTER SPECIAL PROBABILITY DISTRIBUTIONS 90 3.1 Introduction 90 3.2 Special discrete distributions 91 3,3 Special continuous distributions 109 3.4 Location and scale parameters 124 Summary 127 Exercises 128 CHAPTER JOINT DISTRIBUTIONS 136 4.1 Introduction 136 4.2 Joint discrete distributions 137 4.3 Joint continuous distributions 144 4.4 Independent random variables 149 4.5 Conditional distributions 153 4.6 Random samples 158 Summary 165 Exercises 165 CHAPTER PROPERTIES OF RANDOM VARIABLES 171 5.1 Introduction . 171 5.2 Properties of expected values 172 5.3 Correlation 177 5.4 Conditional expectation 180 5;5 Joint moment generating functions 186 Summary 188 Exercises 189 PDF compression, OCR, web optimization using a watermarked evaluation copy of CVISION PDFCompressor FUNCTIONS OF RANDOM VARIABLES 193 6.1 Introduction 193 6.2 The CDF' technique 194 6.3 Transformation methods 197 6.4 Sums of random variables 209 6.5 Order statistics 214 Summary 226 Exercises 226 CHA PTER LIMITING DISTRIBUTIONS 231 7.1 Introduction 231 7.2 Sequences of random variables 232 7.3 The central limit theorem 236 7.4 Approximations for the binomial distribution 240 7.5 Asymptotic normal distributions 243 7.6 Properties of stochastic convergence 245 7.7 Additional limit theorems 247 7.8* Asymptotic distributions of extreme-order statistics 250 Summary 259 Exercises 259 CHAPTER STATISTICS AND SAMPLING DISTRIBUTIONS 263 8.1 Introduction 263 8.2 Statistics 263 8.3 Sampling distributions . 267 8.4 The t, F, and beta distributions 273 8.5 Large-sample approximations 280 Summary 283 Exercises 283 * Advanced (or optional) topics CONTENTS vu CHAPTER PDF compression, OCR, web optimization using a watermarked evaluation copy of CVISION PDFCompressor viii CONTENTS CHAPTER POINT ESTIMATION 288 9.1 Introduction 288 9.2 Some methods of estimation 290 9.3 Criteria for evaluating estimators 302 9.4 Large-sample properties 311 9.5 Bayes and minimax estimators 319 Summary 327 Exercises 328 CHAPTER 10 INTERVAL ESTIMATION 358 11.1 Introduction 358 11.2 Confidence intervals - 359 11.3 Pivotal quantity method 362 11.4 General method 369 11.5 Two-sample problems 377 11.6 Bayesian interval estimation 382 Summary 383 Exercises 384 SUFFICIENCY AND COMPLETENESS 335 10.1 Introduction 335 10.2 Sufficient statistics 337 10.3 Further properties of sufficient statistics 342 10.4 Completeness and exponential class 345 Summary 353 Exercises 353 CHAPTER 11 PDF compression, OCR, web optimization using a watermarked evaluation copy of CVISION PDFCompressor TESTS OF HYPOTHESES 389 12 1 Introduction 389 12.2 Composite hypotheses 395 12.3 Tests for the normal distribution 398 12.4 Binomial tests 404 12.5 Poisson tests 406 12.6 Most powerful tests 406 12.7 Uniformly most powerful tests 411 12.8 Generalized likelihood ratio tests 417 12.9 Conditional tests 426 12.10 Sequential tests 428 Summary 435 Exercises - 436 CHAPTER 13 CONTINGENCY TABLES AND GOODNESS-OF-FIT 442 13.1 Introduction 442 13.2 One-sample binomial case 443 13.3 r-Sample binomial test (completely specified H0) 444 13.4 One-sample multinomial 447 13.5 r-Sample multinomial 448 13.6 Test for independence, r x c contingency table 450 13.7 Chi-squared goodness-of-fit test 453 13.8 Other goodness-of-fit tests 457 Summary 461 Exercises 462 CHAPTER 14 NONPARAMETRIC METHODS 468 14.1 Introduction 468 14.2 One-sample sign test 469 14.3 Binomial test (test on quantiles) 471 14.4 Two-sample sign test 476 CONTENTS ix CI-/APTER 12 PDF compression, OCR, web optimization using a watermarked evaluation copy of CVISION PDFCompressor APPENDIXA REVIEWOFSETS 587 APPENDIX B SPECIAL DISTRIBUTIONS 594 APPENDIX C TABLES OF DISTRIBUTIONS 598 ANSWERS TO SELECTED EXERCISES 619 REFERENCES 638 INDEX 641 Advanced (or optional) topics CONTENTS Wilcoxon paired-sample signed-rank test Paired-sample randomization test Wilcoxon and Mann-Whitney (WMW) tests Correlation teststests of independence Wald-Wolfowjtz runs test Summary Exercises CHAPTER J5* 477 482 483 486 492 494 495 14.5 14.6 14.7 14.8 14.9 15.1 15.2 15.3 15.4 15.5 16.1 16.2 16.3 16.4 16.5 REGRESSION AND LINEAR MODELS Introduction Linear regression Simple linear regression General linear model Analysis of hivariate data Summary Exercises * CHAPTER 499 499 500 501 515 529 534 535 RELIABILITY AND SURVIVAL DISTRIBUTIONS 540 Introduction Reliability concepts Exponential distribution Weibull distribution Repairable systems Summary Exercises 540 541 548 560 570 579 579 PDF compression, OCR, web optimization using a watermarked evaluation copy of CVISION PDFCompressor PREFACE This book provides an introduction to probability and mathematical statistics. Although the primary focus of the book is on a mathematical development of the subject, we also have included numerous examples and exercises that are oriented toward applications. We have attempted to achieve a level of presentation that is appropriate for senior-level undergraduates and beginning graduate students. The second edition involves several major changes, many of which were sug gested by reviewers and users of the first edition. Chapter 2 now is devoted to general properties of random variables and their distributions. The chapter now includes moments and moment generating functions, which occurred somewhat later in the first edition. Special distributions have been placed in Chapter 3. Chapter 8 is completely changed. It now considers sampling distributions and some basic properties of statistics. Chapter 15 is also new. It deals with regression and related aspects of linear models As with the first edition, the only prerequisite for covering the basic material is calculus, with the lone exception of the material on general linear models in Section 15.4; this assumes some familiarity with matrices. This material can be omitted if so desired. Our intent was to produce a book that could be used as a textbook for a two-semester sequence in which the first semester is devoted to probability con- cepts and the second covers mathematfcal statistics. Chapters 1 through 7 include topics that usually are covered in a one-semester introductory course in probabil- ity, while Chapters 8 through 12 contain standard topics in mathematical sta- tistics. Chapters 13 and 14 deal with goodness-of-fit and nonparametric statistics. These chapters tend to be more methods-oriented. Chapters 15 and 16 cover material in regression and reliability, and these would be considered as optional or special topics. In any event, judgment undoubtedly will be required in the xi PDF compression, OCR, web optimization using a watermarked evaluation copy of CVISION PDFCompressor xii PREFACE choice of topics covered or the amount of time allotted to topics if the desired material is to be completed in a two-semester course. lt is our hope that those who use the book will find it both interesting and informative. ACKNOWLEDG M ENTS We gratefully acknowledge the numerous suggestions provided by the following reviewers: Dean H. Fearn Alan M. Johnson Calfornia State UniversityHayward University of Arkansas, Little Rock Joseph Glaz Benny P. Lo University of Connecticut Ohione College Victor Goodman D. Ramachandran Rensselaer Polytechnic Institute California State UniversitySacramento Shu-ping C. Hodgson Douglas A. Wolfe Central Michigan University Ohio State University Robert A. Hultquist Linda J. Young Pennsylvania State University Oklahoma State University Thanks also are due to the following users of the first edition who were kind enough to relate their experiences to the authors: H. A. David, Iowa State Uni- versity; Peter Griffin, California State UniversitySacramento. Finally, special thanks are due for the moral support of our wives, Harriet Bain and Linda Engelhardt. Lee J. Bain Max Engelhardt PDF compression, OCR, web optimization using a watermarked evaluation copy of CVISION PDFCompressor C H A P T 1.1 INTRODUCTION In any scientific study of a physical phenomenon, it is desirable to have a mathe- matical model that makes it possible to describe or predict the observed value of some characteristic of interest. As an example, consider the velocity of a falling body after a certain length of time, t. The formula y = gt, where g 32.17 feet per second per second, provides a useful mathematical model for the velocity, in feet per second, of a body falling from rest in a vacuum. This is an example of a deterministic model. For such a model, carrying out repeated experiments under ideal conditions would result in essentially the same velocity each time, and this would be predicted by the model. On the other hand, such a model may not be adequate when the experiments are carried out under less than ideal conditions. There may be unknown or uncontrolled variables, such as air temperature or humidity, that might affect the outcome, as well as measurement error or other factors that might cause the results to vary on different performances of the PROBtJ3ILITY I PDF compression, OCR, web optimization using a watermarked evaluation copy of CVISION PDFCompressor 2 CHAPTER 1 PROBABILITY experiment. Furthermore, we may not have sufficient knowledge to derive a more complicated model that could account for all causes of variation. There are also other types of phenomena in which different results may natu- rally occur by chance, and for which a deterministic model would not be appro. priate. For example, an experiment may consist of observing the number of particles emitted by a radioactive source, the time until failure of a manufactured component, or the outcome of a game of chance. The motivation for the study of probability is to provide mathematical models for such nondeterministic situations; the corresponding mathematical models will be called probability models (or probabilistic models). The term stochastic, which is derived from the Greek word stochos, meaning "guess," is sometimes used instead of the term probabilistic. A careful study of probability models requires some familiarity with the nota- tion and terminology of set theory. We will assume that the reader has some knowledge of sets, but for convenience we have included a review of the basic ideas of set theory in Appendix A. L2 NOTATION AND TERMINOLOGY The term experiment refers to the process of obtaining an observed result of some phenomenon. A performance of an experiment is called a trial of the experiment, and an observed result is called an outcome. This terminology is rather general, and it could pertain to such diverse activities as scientific experiments or games of chance. Our primary interest will be in situations where there is uncertainty about which outcome will occur when the experiment is performed We will assume that an experiment is repeatable under essentially the same conditions, and that the set of all possible outcomes can be completely specified before experimentation. Definition 1.2.1 The set of all possible outcomes of an experiment is called the sample space, denoted by S. Note Chat one and only one of the possible outcomes will occur on any given trial of the experiments. Exaíipk i .2.1 An experiment consists of tossing two coins, and the observed face of each coin is of interest. The set of possible outcomes may be represented by the sample space S = {HH, HT, TH, TT} PDF compression, OCR, web optimization using a watermarked evaluation copy of CVISION PDFCompressor 1.2 NOTATION AND TERMINOLOGY 3 which simply lists all possible pairings of the symbols H (heads) and T (tails). An alternate way of representing such a sample space is to list all possible ordered pairs of the numbers i and O, S = {(l, 1), (1, 0), (0, 1), (0, O)}, where, for example, (1, 0) indicates that the first coin landed heads up and the second coin landed tails up. Example 1.2.2 Suppose that in Example 1.2.1 we were not interested in the individual outcomes of the coins, but only in the total number of heads obtained from the two coins. An appropriate sample space could then be written as S'1' = {0, 1, 2}. Thus, differ- ent sample spaces may be appropriate for the same experiment, depending on the characteristic of interest. Exampl& 1.2.3 If a coin is tossed repeatedly until a head occurs, then the natural sample space is S = {H, TH, TTH, . . .}. If one is interested in the number of tosses required to obtain a head, then a possible sample space for this experiment would be the set of all positive integers, S'1 = {1, 2, 3, . . .}, and the outcomes would correspond directly to the number of tosses required to obtain the first head. We will show in the next chapter that an outcome corresponding to a sequence of tosses in which a head is never obtained need not be included in the sample space. Exampk 1.2.4 A light bulb is placed in service and the time of operation until it burns out is measured, At least conceptually, the sample space for this experiment can be taken to be the set of nonnegative real numbers, S = {tIO t < c}. Note that if the actual failure time could be measured only to the nearest hour, then the sample space for the actual observed failure time would be the set of nonnegative integers, S'i' = {O, 1, 2, 3, . . .}. Even though S* may be the observable sample space, one might prefer to describe the properties and behavior of light bulbs in terms of the conceptual sample space S. In cases of this type, the dis- creteness imposed by measurement limitations is sufficiently negligible that it can be ignored, and both the measured response and the conceptual response can be discussed relative to the conceptual sample space S. A sample space S is said to be finite if it consists of a finite number of out- comes, say S = {e1, e2 .....eN}, and it is said to be countably infinite if its out- comes can be put into a one-to-one correspondence with the positive integers, say S= {e1,e2,...}. Definition 1.2.2 If a sample space S is either finite or countably infinite then it is called a thscrete sample space. PDF compression, OCR, web optimization using a watermarked evaluation copy of CVISION PDFCompressor

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