Introductory Statistics and Random Phenomena: Uncertainty, Complexity and Chaotic Behavior in Engineering and Science PDF
Preview Introductory Statistics and Random Phenomena: Uncertainty, Complexity and Chaotic Behavior in Engineering and Science
Errata Introductory Statistics and Random Phenomena Manfred Denker • Wojbor A. Woyczynski Page 498: Heading shouldread: Tables: Standard Normal CumulativeProbabilities <I>(z) Page499: Heading shouldread: Tables: T-distribution, UpperTail Quantilesta(n) Introductory Statistics and Random Phenomena Uncertainty, Complexity and Chaotic Behavior in Engineering and Science with Mathematica® Uncertain Virtual Worlds™ by Bernard Y cart Manfred Denker Wojbor A. Woyczyriski Springer Science+Business Media, LLC Manfred Denker W ojbor A. W oyczynski Georg-August-Universităt Case Westem Reserve University Gottingen, GERMANY Cleveland, OH 8emard Y cart Universite Joseph Fourier Grenoble, FRANCE Library of Congress Cataloging-in-Publication Data Denker, Manfred, 1944- Introductory statistics and random phenomena : uncertainty, complexity, and chaotic behavior in engineering and science / M. Denker and W.A. Woyczynski ; with Mathematica Uncertain Virtual Worlds by B. Ycart. p. cm. --(Statistics for industry and technology) Includes bibliographical references and index. ISBN 978-1-4612-7388-2 ISBN 978-1-4612-2028-2 (eBook) DOI 10.1007/978-1-4612-2028-2 1. Mathematical statistics. 2. Probabilities. 3. Mathematica (computer file) 1. Woyczynski, W. A. (Wojbor Andrzej) , 1943- II. Ycart, B. (Bernard), 1958- . III. Title. IV. Series. QA276.12.D45 1998 519.5--dc21 98-4735 CIP m Printed on acid-free paper ® © 1998 Springer Science+Business Media New York a{j?) Originally published by Birkhiiuser Boston in 1998 Softcover reprint of the hardcover 1s t edition 1998 Copyright is not c1aimed for works of U .S. Government employees. AII rights reserved. No part ofthis publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, record ing, or otherwise, without prior permission ofthe copyright owner. Authorization to photocopy items for internal or personal use, or the internal or personal use of specific ci ients, is granted by Springer Science+Business Media, LLC, provided that the appropriate fee is paid directly to Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, USA (Telephone: (978) 750-8400), stating the ISBN, the title ofthe book, and the first and last page numbers of each article copied. The copyright owner's consent does not include copying for general distribution, promotion, new works, or resale. In these cases, specific written permission must first be obtained from the publisher. ISBN 978-1-4612-7388-2 Mathematica' is a registered trademark ofWolfram, Inc., 100 Trade Center Drive, Champaign, IL 61820-7237 Typeset by T & T Techworks Inc., Coral Springs, FL 9 8 7 6 5 4 3 2 1 Contents Preface xi Introduction xv NotationandAbbreviations xxiii I DESCRIPTIVE STATISTICS-COMPRESSING DATA 1 1 WhyOneNeedstoAnalyzeData 3 1.1 Cointossing, lottery, andthestockmarket 3 1.2 Inventoryproblemsinmanagement . . . 9 1.3 Batterylifeandqualitycontrolin manufacturing. 10 1.4 Reliabilityofcomplexsystems. . 12 1.5 Pointprocessesintimeandspace 17 1.6 Polls-socialsciences . . . . . . . 21 1.7 Timeseries . . . . . . . . . . . . 26 1.8 Repeatedexperimentsandtesting 29 1.9 Simplechaoticdynamicalsystems 32 1.10 Complexdynamicalsystems . . . . 39 1.11 PseudorandomnumbergeneratorsandtheMonte-Carlomethods 41 1.12 Fractalsandimagereconstruction . . . . . 45 1.13 Codinganddecoding, unbreakableciphers. 46 1.14 Experiments,exercises,andprojects 49 1.15 Bibliographicalnotes. . . . . . . . 52 2 DataRepresentationandCompression 55 2.1 Datatypes,categoricaldata . . . . . . . . . . . . 55 2.2 Numericaldata: orderstatistics,median,quantiles . 63 2.3 Numericaldata: histograms, means, moments 70 2.4 Location,dispersion,andshapeparameters .... 77 v vi Contents 2.5 Probabilities: afrequentistviewpoint 82 2.6 Multidimensionaldata: histograms andothergraphicalrepresen- tations . . . . . . . . . . . . . . . . 88 2.7 2-Ddata: regressionandcorrelations . 92 2.8 Fractaldata. . . . . . . . . . . . . . . 99 2.9 Measuringinformationcontententropy 105 2.10 Experiments,exercises,andprojects 111 2.11 Bibliographicalnotes. . . . . . . . . . 115 3 AnalyticRepresentationofRandomExperimentalData 119 3.1 Repeatedexperimentsandthelawoflargenumbers . . . . . .. 119 3.2 Characteristics ofexperiments: distribution functions, densities, means,variances 128 3.3 Uniformdistributions,simulationofrandomquantities,theMonte Carlomethod. . . . . . . . . . . . . . . . . . 136 3.4 Bernoulliandbinomialdistributions . . . . . . . . . . 139 3.5 Rescalingprobabilities: Poissonapproximation . . . . 145 3.6 StabilityofFluctuationsLaw: Gaussianapproximation 152 3.7 Howtoestimatep inBernoulliexperiments . . . . . . 163 3.8 Othercontinuousdistributions;Gammafunction calculus 171 3.9 Testingthefitofadistribution . . . . . . . . 185 3.10 Randomvectorsandmultivariatedistributions 188 3.11 Experiments,exercises, andprojects 196 3.12 Bibliographicalnotes. . . . . . . . . . . . . 198 II MODELING UNCERTAINTY 201 4 AlgorithmicComplexityandRandomStrings 203 4.1 Heartofrandomness: whenisrandom- random? . 203 4.2 ComputablestringsandtheTuringmachine . . . . 207 4.3 Ko1mogorovcomplexityandrandomstrings. . . . 212 4.4 Typical sequences: Martin-Loftestsofrandomness 218 4.5 Stabilityofsubsequences: vonMisesrandomness . 226 4.6 Computableframeworkofrandomness: degreesofirregularity 228 4.7 Experiments,exercises,andprojects 237 4.8 Bibliographicalnotes. . . . . . . . . . . . . . . . . . . . . 240 5 StatisticalIndependenceand Kolmogorov'sProbabilityTheory 243 5.1 Descriptionofexperiments,randomvariables,andKolmogorov's axioms. . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 5.2 Uniformdiscretedistributionsandcounting . . . . . . . . . 258 5.3 Statisticalindependenceasamodelforrepeatedexperiments 261 5.4 Expectationsandothercharacteristicsofrandomvariables. 265 Contents vii 5.4.1 Expectations. 265 5.4.2 Expectationsoffunctions ofrandomvariables. Variance. 267 5.4.3 Expectationsoffunctionsofvectors. Covariance. . . .. 268 5.4.4 Expectationofthe product. Variance ofthe sumofinde- pendentrandomvariables. . . . . . . . . . . . 269 5.4.5 Momentsandthemomentgeneratingfunction. 272 5.4.6 Expectationsofgeneralrandomvariables. . 275 5.5 Averagesofindependentrandomvariables. 276 5.6 Lawsoflargenumbersandsmalldeviations 281 5.7 Centrallimittheoremandlargedeviations 284 5.8 Experiments,exercises,andprojects 287 5.9 BibliographicalNotes . . . . . . . . . . 292 6 Chaos in Dynamical Systems: How Uncertainty Arises in Scientific andEngineeringPhenomena 293 6.1 Dynamicalsystems: generalconceptsandtypicalexamples 293 6.2 Orbitsandfixedpoints . . . . . . . . . . . . . . . . 307 6.3 Stabilityoffrequencies andtheergodictheorem . . . 325 6.4 Stabilityoffluctuations andthecentrallimittheorem 340 6.5 Attractors,fractals, andentropy . . 349 6.6 Experiments,exercises, andprojects 360 6.7 Bibliographicalnotes . 362 III MODEL SPECIFICATION-DESIGN OF EXPERIMENTS 365 7 GeneralPrinciplesofStatisticalAnalysis 367 7.1 Designofexperimentsandplanningofinvestigation . 367 7.2 Modelselection .... . . . . . . . . . . . . 369 7.3 Determiningthemethodofstatisticalinference 372 7.3.1 Maximumlikelihoodestimator(MLE) . 373 7.3.2 Leastsquaresestimator(LSE) 376 7.3.3 Methodofmoments(MM) . 381 7.3.4 Concludingremarks ..... 383 7.4 Estimationoffractal dimension . . . 384 7.5 Practicalsideofdatacollectionandanalysis . 387 7.6 Experiments,exercises, andprojects 389 7.7 Bibliographicalnotes . 390 8 StatisticalInferenceforNormalPopulations 393 8.1 Introduction; parametricinference . . . . 393 8.2 Confidenceintervalsforone-samplemodel .. 406 8.3 Fromconfidenceintervalstohypothesis testing 414 viii Contents 8.4 Statisticalinferencefortwo-samplenormalmodels 422 8.5 Regressionanalysisforthenormalmodel 427 8.6 Testingforgoodness-of-fit . . . . . 435 8.7 Experiments,exercises,andprojects 438 8.8 Bibliographicalnotes. . . . . . . . 440 9 AnalysisofVariance 443 9.1 Single-factorANOVA . 443 9.2 Two-factorANOVA . 448 9.3 Experiments,exercises,andprojects 456 9.4 Bibliographicalnotes. . . . . . . . 459 A UncertaintyPrincipleinSignalProcessingandQuantumMechanics 461 B FuzzySystemsandLogic 465 C ACritiqueofPureReason 469 D TheRemarkableBernoulliFamily 473 E UncertainVirtualWorldsMathematica Packages 477 F Tables 497 Index 503 Toourwives, JEANNE,LIZ,andTILLU Preface ThepresentbookisbasedonacoursedevelopedaspartofthelargeNSF-funded GatewayCoalitionInitiativeinEngineeringEducationwhichincludedCaseWest ern Reserve University, Columbia University, CooperUnion, Drexel University, Florida International University, New Jersey Institute ofTechnology, Ohio State University, UniversityofPennsylvania,PolytechnicUniversity,andUniversityof South Carolina. The Coalition aimed to restructure the engineering curriculum byincorporatingthelatesttechnologicalinnovationsandtriedtoattractmoreand betterstudentstoengineeringandscience. Draftsofthistextbookhavebeenused since1992instatisticscoursestaughtatCWRU,IndianaUniversity,Bloomington, andattheuniversitiesinGottingen,Germany,andGrenoble, France. Another purpose ofthis project was to develop a courseware that would take advantageoftheElectronicLearningEnvironmentcreatedbyCWRUnet-theall fiber-optic CaseWesternReserve Universitycomputernetwork, and itsability to letstudentsrunMathematica experiments and projects in theirdormitoryrooms, andinteractpaperlesslywiththe instructor. Theoretically,onecouldtrytogothroughthisbookwithoutdoingMathematica experimentsonthecomputer,butitwouldbelikeplayingChopin'sPianoConcerto in E-minor, orPink Floyd's The Wall, on an accordion. One would getan idea ofwhatthetunewaswithouteverexperiencingthefullrichnessandpowerofthe entirecomposition, andthewholeambience wouldbemiscued. Acknowledgments Thanksareduetoseveralgroupsofstudentsthathavetakendifferentversionsof thiscourseoverthelastsixyears. Theypatientlyandconsistentlyfoundmistakes, produced good graphics, and came up with interesting data sets from their own disciplines. Theirindividualcontributions areacknowledgedin thetext. Wealso appreciatehelpfromTom RyanandJiming JiangoftheCWRUStatisticsDepart ment,whoreadvariousportionsofthemanuscriptandpointedoutplacesthatcould be improved upon. We thank Steve Pinkus ofYale University and Burt Singer ofPrinceton University for discussing with us their recent work on computable xi
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