Probability and Statistics

F.M. Dekking C. Kraaikamp H.P. Lopuhaä L.E. Meester A Modern Introduction to Probability and Statistics Understanding Why and How With 120 Figures Frederik Michel Dekking Cornelis Kraaikamp Hendrik Paul Lopuhaä Ludolf Erwin Meester Delft Institute of Applied Mathematics Delft University of Technology Mekelweg 4 2628 CD Delft The Netherlands Whilst we have made considerable efforts to contact all holders of copyright material contained in this book, we may have failed to locate some of them. Should holders wish to contact the Publisher, we will be happy to come to some arrangement with them. British Library Cataloguing in Publication Data A modern introduction to probability and statistics. — (Springer texts in statistics) 1. Probabilities 2. Mathematical statistics I. Dekking, F. M. 519.2 ISBN 1852338962 Library of Congress Cataloging-in-Publication Data A modern introduction to probability and statistics : understanding why and how / F.M. Dekking … [et al.]. p. cm. — (Springer texts in statistics) Includes bibliographical references and index. ISBN 1-85233-896-2 1. Probabilities—Textbooks. 2. Mathematical statistics—Textbooks. I. Dekking, F.M. II. Series. QA273.M645 2005 519.2—dc22 2004057700 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. ISBN-10: 1-85233-896-2 ISBN-13: 978-1-85233-896-1 Springer Science+Business Media springeronline.com © Springer-Verlag London Limited 2005 The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Printed in the United States of America 12/3830/543210 Printed on acid-free paper SPIN 10943403 Preface Probability and statistics are fascinating subjects on the interface between mathematics and applied sciences that help us understand and solve practical problems. We believe that you, by learning how stochastic methods come about and why they work, will be able to understand the meaning of statistical statements as well as judge the quality of their content, when facing such problems on your own. Our philosophy is one of how and why: instead of just presenting stochastic methods as cookbook recipes, we prefer to explain the principles behind them. In this book you will find the basics of probability theory and statistics. In addition, there are several topics that go somewhat beyond the basics but that ought to be present in an introductory course: simulation, the Poisson process, the law of large numbers, and the central limit theorem. Computers have brought many changes in statistics. In particular, the bootstrap has earned its place. It provides the possibility to derive confidence intervals and perform tests of hypotheses where traditional (normal approximation or large sample) methods are inappropriate. It is a modern useful tool one should learn about, we believe. Examples and datasets in this book are mostly from real-life situations, at least that is what we looked for in illustrations of the material. Anybody who has inspected datasets with the purpose of using them as elementary examples knows that this is hard: on the one hand, you do not want to boldly state assumptions that are clearly not satisfied; on the other hand, long explanations concerning side issues distract from the main points. We hope that we found a good middle way. A first course in calculus is needed as a prerequisite for this book. In addition to high-school algebra, some infinite series are used (exponential, geometric). Integration and differentiation are the most important skills, mainly concerning one variable (the exceptions, two dimensional integrals, are encountered in Chapters 9–11). Although the mathematics is kept to a minimum, we strived VI Preface to be mathematically correct throughout the book. With respect to probability and statistics the book is self-contained. The book is aimed at undergraduate engineering students, and students from more business-oriented studies (who may gloss over some of the more mathematically oriented parts). At our own university we also use it for students in applied mathematics (where we put a little more emphasis on the math and add topics like combinatorics, conditional expectations, and generating functions). It is designed for a one-semester course: on average two hours in class per chapter, the first for a lecture, the second doing exercises. The material is also well-suited for self-study, as we know from experience. We have divided attention about evenly between probability and statistics. The very first chapter is a sampler with differently flavored introductory examples, ranging from scientific success stories to a controversial puzzle. Topics that follow are elementary probability theory, simulation, joint distributions, the law of large numbers, the central limit theorem, statistical modeling (informal: why and how we can draw inference from data), data analysis, the bootstrap, estimation, simple linear regression, confidence intervals, and hypothesis testing. Instead of a few chapters with a long list of discrete and continuous distributions, with an enumeration of the important attributes of each, we introduce a few distributions when presenting the concepts and the others where they arise (more) naturally. A list of distributions and their characteristics is found in Appendix A. With the exception of the first one, chapters in this book consist of three main parts. First, about four sections discussing new material, interspersed with a handful of so-called Quick exercises. Working these—two-or-three-minute— exercises should help to master the material and provide a break from reading to do something more active. On about two dozen occasions you will find indented paragraphs labeled Remark, where we felt the need to discuss more mathematical details or background material. These remarks can be skipped without loss of continuity; in most cases they require a bit more mathematical maturity. Whenever persons are introduced in examples we have determined their sex by looking at the chapter number and applying the rule “He is odd, she is even.” Solutions to the quick exercises are found in the second to last section of each chapter. The last section of each chapter is devoted to exercises, on average thirteen per chapter. For about half of the exercises, answers are given in Appendix C, and for half of these, full solutions in Appendix D. Exercises with both a short answer and a full solution are marked with  and those with only a short answer are marked with  (when more appropriate, for example, in “Show that . . . ” exercises, the short answer provides a hint to the key step). Typically, the section starts with some easy exercises and the order of the material in the chapter is more or less respected. More challenging exercises are found at the end. Preface VII Much of the material in this book would benefit from illustration with a computer using statistical software. A complete course should also involve computer exercises. Topics like simulation, the law of large numbers, the central limit theorem, and the bootstrap loudly call for this kind of experience. For this purpose, all the datasets discussed in the book are available at http://www.springeronline.com/1-85233-896-2. The same Web site also provides access, for instructors, to a complete set of solutions to the exercises; go to the Springer online catalog or contact textbooks@springer-sbm.com to apply for your password. Delft, The Netherlands January 2005 F. M. Dekking C. Kraaikamp H. P. Lopuhaä L. E. Meester Contents 1 2 Why probability and statistics? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Biometry: iris recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1.2 Killer football . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Cars and goats: the Monty Hall dilemma . . . . . . . . . . . . . . . . . . . 1.4 The space shuttle Challenger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 5 1.5 Statistics versus intelligence agencies . . . . . . . . . . . . . . . . . . . . . . . 7 1.6 The speed of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Outcomes, events, and probability . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1 Sample spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 Products of sample spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.5 An infinite sample space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.6 Solutions to the quick exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3 Conditional probability and independence . . . . . . . . . . . . . . . . . 25 3.1 Conditional probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 The multiplication rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.3 The law of total probability and Bayes’ rule . . . . . . . . . . . . . . . . . 30 3.4 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.5 Solutions to the quick exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 X Contents 4 Discrete random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The probability distribution of a discrete random variable . . . . 4.3 The Bernoulli and binomial distributions . . . . . . . . . . . . . . . . . . . 4.4 The geometric distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Solutions to the quick exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 41 43 45 48 50 51 5 Continuous random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Probability density functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The uniform distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The exponential distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 The Pareto distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 The normal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Quantiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Solutions to the quick exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 57 60 61 63 64 65 67 68 6 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 What is simulation? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Generating realizations of random variables . . . . . . . . . . . . . . . . . 6.3 Comparing two jury rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 The single-server queue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Solutions to the quick exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 71 72 75 80 84 85 7 Expectation and variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Expected values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Three examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 The change-of-variable formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Solutions to the quick exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 89 93 94 96 99 99 8 Computations with random variables . . . . . . . . . . . . . . . . . . . . . . 103 8.1 Transforming discrete random variables . . . . . . . . . . . . . . . . . . . . 103 8.2 Transforming continuous random variables . . . . . . . . . . . . . . . . . . 104 8.3 Jensen’s inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Contents XI 8.4 Extremes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 8.5 Solutions to the quick exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 8.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 9 Joint distributions and independence . . . . . . . . . . . . . . . . . . . . . . 115 9.1 Joint distributions of discrete random variables . . . . . . . . . . . . . . 115 9.2 Joint distributions of continuous random variables . . . . . . . . . . . 118 9.3 More than two random variables . . . . . . . . . . . . . . . . . . . . . . . . . . 122 9.4 Independent random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 9.5 Propagation of independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 9.6 Solutions to the quick exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 9.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 10 Covariance and correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 10.1 Expectation and joint distributions . . . . . . . . . . . . . . . . . . . . . . . . 135 10.2 Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 10.3 The correlation coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 10.4 Solutions to the quick exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 10.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 11 More computations with more random variables . . . . . . . . . . . 151 11.1 Sums of discrete random variables . . . . . . . . . . . . . . . . . . . . . . . . . 151 11.2 Sums of continuous random variables . . . . . . . . . . . . . . . . . . . . . . 154 11.3 Product and quotient of two random variables . . . . . . . . . . . . . . 159 11.4 Solutions to the quick exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 11.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 12 The Poisson process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 12.1 Random points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 12.2 Taking a closer look at random arrivals . . . . . . . . . . . . . . . . . . . . . 168 12.3 The one-dimensional Poisson process . . . . . . . . . . . . . . . . . . . . . . . 171 12.4 Higher-dimensional Poisson processes . . . . . . . . . . . . . . . . . . . . . . 173 12.5 Solutions to the quick exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 12.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 13 The law of large numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 13.1 Averages vary less . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 13.2 Chebyshev’s inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 XII Contents 13.3 The law of large numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 13.4 Consequences of the law of large numbers . . . . . . . . . . . . . . . . . . 188 13.5 Solutions to the quick exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 13.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 14 The central limit theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 14.1 Standardizing averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 14.2 Applications of the central limit theorem . . . . . . . . . . . . . . . . . . . 199 14.3 Solutions to the quick exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 14.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 15 Exploratory data analysis: graphical summaries . . . . . . . . . . . . 207 15.1 Example: the Old Faithful data . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 15.2 Histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 15.3 Kernel density estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 15.4 The empirical distribution function . . . . . . . . . . . . . . . . . . . . . . . . 219 15.5 Scatterplot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 15.6 Solutions to the quick exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 15.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 16 Exploratory data analysis: numerical summaries . . . . . . . . . . . 231 16.1 The center of a dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 16.2 The amount of variability of a dataset . . . . . . . . . . . . . . . . . . . . . . 233 16.3 Empirical quantiles, quartiles, and the IQR . . . . . . . . . . . . . . . . . 234 16.4 The box-and-whisker plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 16.5 Solutions to the quick exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 16.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 17 Basic statistical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 17.1 Random samples and statistical models . . . . . . . . . . . . . . . . . . . . 245 17.2 Distribution features and sample statistics . . . . . . . . . . . . . . . . . . 248 17.3 Estimating features of the “true” distribution . . . . . . . . . . . . . . . 253…
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