Probability and Statistics in Experimental Physics PDF
Preview Probability and Statistics in Experimental Physics
UNDERGRADUATE TEXTS IN CONTEMPORARY PHYSICS Editors Robert S. Averback Robert C. Hilborn David Peak Thomas Rossing Cindy Schwarz Springer New York Berlin Heidelberg Barcelona Hong Kong London Milan Paris Singapore Tokyo UNDERGRADUATE TEXTS IN CONTEMPORARY PHYSICS Enns and McGuire, Computer Algebra Recipes: A Gourmet's Guide to the Mathematical Models of Science Hassani, Mathematical Methods: For Students of Physics and Related Fields Holbrow, Lloyd, and Amato, Modern Introductory Physics Roe, Probability and Statistics in Experimental Physics, Second Edition Rossing and Chiaverina, Light Science: Physics and the Visual Arts PROBABILITY AND STATISTICS IN EXPERIMENTAL PHYSICS Second Edition Byron P. Roe With 44 Illustrations i Springer Byron P. Roe Department of Physics Randall Laboratory University of Michigan Ann Arbor, MI 49109 USA Series Editors Robert S. Averback Robert C. Hilborn Department of Materials Science Department of Physics University of Illinois Amherst College Urbana, IL 61801 Amherst, MA OlD02 USA USA David Peak Thomas D. Rossing Department of Physics Department of Physics Utah State University Northern illinois University Logan, UT 84322 De Kalb, IL 60115-2854 USA USA Cindy Schwartz Department of Physics and Astronomy Vassar College Poughkeepsie, NY 12601 USA Library of Congress Cataloging-in-Publication Data Roe, Byron P. Probability and statistics in experimental physics / Byron P. Roe.-2nd ed. p. ctn. - (Undergraduate texts in contemporary physics) Includes bibliographical references and index. ISBN 978-1-4419-2895-5 ISBN 978-1-4684-9296-5 (eBook) DOl 10.1007/978-1-4684-9296-5 1. Physics-Experiments-Technique. 2. Statistical physics. 2. Probabilities. I. Title. II. Series. QCH .R59 2001 530 .078-dc21 00-047098 Printed on acid-free paper. © 2001 Springer-Verlag New York, Inc. Softcover reprint of the hardcover 2nd edition 2001 All rights reserved. This work may not be translated or copied in whole or in pan without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews orscholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimi lar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Production managed by Timothy Taylor; manuf~cturing supervised by Jeffrey Taub. Cameracready copy provided by the author. 9 8 7 6 5 4 3 2 SPIN 10785204 Springer-Verlag New York Berlin Heidelberg A member of BertelsmannSpringer Science+Business Media GmbH Preface This book is meant to be a practical introduction into the use of probability and statistics in experimental physics for advanced undergraduate students and for graduate students. I have attempted to write a short book. It is not intended as a comprehensive text in probability and statistics. I have tried to emphasize areas I have found to be useful when doing experimental physics. Except for the first two chapters the emphasis is on applications and understanding. I have omitted proofs of formal theorems in the interests of brevity unless I felt the proof added to one's intuition in understanding and applying the theorem. Since, however, this is a field in which there are often a number of misunderstandings, it is necessary to state some things with reasonable precision. I have tried to do this when necessary. I assume the student is familiar with partial derivatives and with ele mentary matrix manipulation. A computer is a needed tool for probability and statistics in experimental physics. We will introduce its use in this subject in some of the homework exercises. One may interact with a computer in a batch mode or an inter active mode. In a batch mode, one submits FORTRAN or other language programs, the computer processes them, and returns the end results. In the interactive mode, one gives the computer an instruction, the computer processes it, indicates what it has done, and waits for the next instruction. In the homework exercises in this book, random number routines, his togram routines, minimizing routines, and matrix manipulation routines will be needed. In batch mode, the CERN library, available in many physics depart ments, provides excellent packages in all these areas and is highly rec ommended. If the CERN library is not available, the very useful book Numerical Recipes contains programs for pseudo-random numbers, for [IJ minimizing functions, and for matrix manipulation. These programs are available on a diskette. In the interactive mode, the MAPLE software system available commer cially can provide most of the packages needed. The SAS system or the PCSAS system also provides most of the needed functionality. Any of these systems will require an initial learning effort. However, the reward in terms of problem-solving capability is very great and the investment in time well spent. In some of the exercises I will provide some hints on the use of the CERN system. However, the interactive systems may well dominate in the future and they are equally recommended. A solutions manual is now available. vi Preface For this Second Edition, I have made a number of changes and additions. Among the most important of these are a new chapter on queueing theory and a discussion of the Feldman-Cousins unified method for estimating confidence intervals. Expositions have been added on the fitting of weighted events, the fitting of events with errors in x and y, and a number of other topics. Byron P. Roe Contents Preface................................................... v 1. Basic Probability Concepts ............................ . 1 2. Some Initial Definitions ................................ . 5 2.1 Worked Problems .................................... 10 2.2 Exercises. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . 13 3. Some Results Independent of Specific Distributions ............................................ 15 3.1 Multiple Scattering and the Root N Law............. 15 3.2 Propagation of Errors; Errors When Changing Variables ............................................ 19 3.3 Some Useful Inequalities ............................. 21 3.4 Worked Problems .................................... 23 3.5 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4. Discrete Distributions and Combinatorials ........... 29 4.1 Worked Problems.................................... 32 4.2 Exercises... . . . . . . . . .. . . . . . . .. . . . . . . . . . . .. . . . . . . . . . .. 33 5. Specific Discrete Distributions ......................... 35 5.1 Binomial Distribution. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 35 5.2 Poisson Distribution ................................. 37 5.3 Worked Problems .................................... 39 5.4 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 42 6. The Normal (or Gaussian) Distribution and Other Continuous Distributions ....................... 44 6.1 The Normal Distribution............................. 44 6.2 The Chi-square Distribution ......................... 47 6.3 F Distribution ....................................... 49 6.4 Student's Distribution ............................... 51 6.5 The Uniform Distribution............................ 51 6.6 The Log-Normal Distribution ........................ 52 Contents Vlll 6.7 The Cauchy Distribution (Breit-Wigner Distribution) ........................................ 53 6.8 Worked Problems .................................... 55 6.9 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 7. Generating Functions and Characteristic Functions ................................................ 58 7.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . .. . . . . . 58 7.2 Convolutions and Compound Probability ............. 58 7.3 Generating Functions ................................ 60 7.4 Characteristic Functions ............................. 62 7.5 Exercises. . . . . .. . . .. .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .. 65 8. The Monte Carlo Method: Computer Simulation of Experiments ............................. 66 8.1 Using the Distribution Inverse ....................... 66 8.2 Method of Composition .............................. 67 8.3 Acceptance Rejection Method ........................ 69 8.4 Computer Pseudorandom Number Generators........ 70 8.5 Unusual Application of a Pseudorandom Number String....................................... 73 8.6 Worked Problems .................................... 75 8.7 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 9. Queueing Theory and Other Probability Questions ................................................ 81 9.1 Queueing Theory .................................... 81 9.2 Markov Chains................................ ...... 84 9.3 Games of Chance .................................... 87 9.4 Gambler's Ruin ...................................... 89 9.5 Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 10. Two-Dimensional and Multidimensional Distributions ............................................ 92 10.1 Introduction ........................ ,................ 92 10.2 Two-Dimensional Distributions ...................... 92 Contents IX 10.3 Multidimensional Distributions ................ " . . . . . 99 10.4 Theorems on Sums of Squares........................ 103 10.5 Exercises ............................................ 105 11. The Central Limit Theorem............................ 107 11.1 Introduction; Lindeberg Criterion .................... 107 11.2 Failures of the Central Limit Theorem ............... 110 11.3 Khintchine's Law of the Iterated Logarithm .......... 115 11.4 Worked Problems.................................... 116 11.5 Exercises ............................................ 118 12. Inverse Probability; Confidence Limits ................ 119 12.1 Bayes' Theorem ..................................... 119 12.2 The Problem of A Priori Probability ................ 120 12.3 Confidence Intervals and Their Interpretation 122 12.4 Use of Confidence Intervals for Discrete Distributions ........................................ 128 12.5 Improving on the Symmetric Tails Confidence Limits............................................... 131 12.6 When Is a Signal Significant? ........................ 138 12.7 Worked Problems .................................... 141 12.8 Exercises ............................................ 145 13. Methods for Estimating Parameters. Least Squares and Maximum Likelihood..................... 146 13.1 Method of Least Squares (Regression Analysis) ....... 146 13.2 Maximum Likelihood Method........................ 148 13.3 Further Considerations in Fitting Histograms ......... 156 13.4 Improvement over Symmetric Tails Confidence Limits for Events With Partial Background-Signal Separation ....................... 158 13.5 Estimation of a Correlation Coefficient ............... 162 13.6 Putting Together Several Probability Estimates ...... 162 13.7 Worked Problems .................................... 164 13.8 Exercises ............................................ 167 x Contents 14. Curve Fitting 170 14.1 The Maximum Likelihood Method for Multiparameter Problems ..................... " . . . . . 170 14.2 Regression Analysis with Non-constant Variance. . . . . . 173 14.3 The Gibb's Phenomenon ............................. 178 14.4 The Regularization Method.......................... 180 14.5 Other Regularization Schemes........................ 182 14.6 Fitting Data With Errors in Both x and y ............ 184 14.7 Non-linear Parameters............................... 186 14.8 Optimizing a Data Set With Signal and Background ......................................... 191 14.9 Robustness of Estimates ............................. 194 14.10Worked Problems .................................... 195 14.11Exercises ............................................ 198 15. Bartlett S Function; Estimating Likelihood Ratios Needed for an Experiment...................... 201 15.1 Introduction ......................................... 201 15.2 The Jacknife ......................................... 201 15.3 Making the Distribution Function of the Estimate Close to Normal; the Bartlett S Function ............................................ 202 15.4 Likelihood Ratio ..................... , ........ " . . . . . 208 15.5 Estimating in Advance the Number of Events Needed for an Experiment ........................... 209 15.6 Exercises ............................................ 212 16. Interpolating Functions and Unfolding Problems ................................................. 213 16.1 Interpolating Functions .............................. 213 16.2 Spline Functions ..................................... 213 16.3 B-Splines ............................................ 215 16.4 Unfolding Data ...................................... 216 16.5 Exercises ............................................ 219
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