Introductory Statistics and Random Phenomena: Uncertainty, Complexity and Chaotic Behavior in Engineering and Science

519 Pages·1998·30.714 MB·English

by  Manfred Denker, Wojbor A. Woyczyński, Bernard Ycart (auth.)

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
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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