Introduction to Probability and Statistics Using R

386 Pages·2010·2.31 MB·English

by  G. Jay Kerns

Introduction to Probability
and Statistics Using R
G. Jay Kerns
irst dition
F E
ii
IPUR: IntroductiontoProbabilityand StatisticsUsingR
S
Copyright© 2010G. JayKerns
ISBN: 978-0-557-24979-4
Permission is granted to copy, distribute and/or modify this document under the terms of the
GNU Free Documentation License, Version 1.3 or any later version published by the Free
Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover
Texts. A copy of the license is included in the section entitled “GNU Free Documentation
License”.
Date: July28,2010
Contents
Preface vii
ListofFigures xiii
ListofTables xv
1 An Introduction to Probability andStatistics 1
1.1 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 An Introduction to R 5
2.1 Downloadingand InstallingR . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 CommunicatingwithR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 BasicROperationsand Concepts . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 GettingHelp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5 ExternalResources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.6 OtherTips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 Data Description 19
3.1 TypesofData . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Features ofDataDistributions . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3 DescriptiveStatistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.4 ExploratoryDataAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.5 MultivariateDataand DataFrames . . . . . . . . . . . . . . . . . . . . . . . . 45
3.6 ComparingPopulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4 Probability 65
4.1 SampleSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2 Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.3 ModelAssignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.4 Properties ofProbability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.5 CountingMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.6 ConditionalProbability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.7 IndependentEvents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.8 Bayes’Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.9 RandomVariables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
iii
iv CONTENTS
5 Discrete Distributions 107
5.1 DiscreteRandom Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.2 TheDiscreteUniformDistribution . . . . . . . . . . . . . . . . . . . . . . . . 110
5.3 TheBinomialDistribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.4 Expectationand MomentGeneratingFunctions . . . . . . . . . . . . . . . . . 116
5.5 TheEmpiricalDistribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.6 OtherDiscreteDistributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.7 FunctionsofDiscreteRandom Variables . . . . . . . . . . . . . . . . . . . . . 130
Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6 Continuous Distributions 137
6.1 ContinuousRandom Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.2 TheContinuousUniformDistribution . . . . . . . . . . . . . . . . . . . . . . 142
6.3 TheNormal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.4 FunctionsofContinuousRandomVariables . . . . . . . . . . . . . . . . . . . 146
6.5 OtherContinuousDistributions . . . . . . . . . . . . . . . . . . . . . . . . . . 150
Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
7 MultivariateDistributions 157
7.1 Jointand MarginalProbabilityDistributions . . . . . . . . . . . . . . . . . . . 157
7.2 Jointand MarginalExpectation . . . . . . . . . . . . . . . . . . . . . . . . . . 163
7.3 ConditionalDistributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
7.4 IndependentRandom Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 167
7.5 ExchangeableRandom Variables . . . . . . . . . . . . . . . . . . . . . . . . . 170
7.6 TheBivariateNormalDistribution . . . . . . . . . . . . . . . . . . . . . . . . 170
7.7 BivariateTransformationsofRandomVariables . . . . . . . . . . . . . . . . . 172
7.8 Remarks fortheMultivariateCase . . . . . . . . . . . . . . . . . . . . . . . . 175
7.9 TheMultinomialDistribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
8 Sampling Distributions 181
8.1 SimpleRandomSamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
8.2 SamplingfromaNormalDistribution . . . . . . . . . . . . . . . . . . . . . . 182
8.3 TheCentral LimitTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
8.4 SamplingDistributionsofTwo-SampleStatistics . . . . . . . . . . . . . . . . 187
8.5 SimulatedSamplingDistributions . . . . . . . . . . . . . . . . . . . . . . . . 189
Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
9 Estimation 193
9.1 PointEstimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
9.2 ConfidenceIntervalsforMeans . . . . . . . . . . . . . . . . . . . . . . . . . . 202
9.3 ConfidenceIntervalsforDifferences ofMeans . . . . . . . . . . . . . . . . . . 208
9.4 ConfidenceIntervalsforProportions . . . . . . . . . . . . . . . . . . . . . . . 210
9.5 ConfidenceIntervalsforVariances . . . . . . . . . . . . . . . . . . . . . . . . 212
9.6 FittingDistributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
9.7 SampleSize and MarginofError . . . . . . . . . . . . . . . . . . . . . . . . . 212
9.8 OtherTopics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
CONTENTS v
10 Hypothesis Testing 217
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
10.2 TestsforProportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
10.3 OneSampleTests forMeansand Variances . . . . . . . . . . . . . . . . . . . 224
10.4 Two-SampleTestsfor Meansand Variances . . . . . . . . . . . . . . . . . . . 227
10.5 OtherHypothesisTests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
10.6 AnalysisofVariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
10.7 SampleSize and Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
11 Simple LinearRegression 235
11.1 BasicPhilosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
11.2 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
11.3 ModelUtilityand Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
11.4 ResidualAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
11.5 OtherDiagnosticTools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
12 Multiple LinearRegression 267
12.1 TheMultipleLinearRegressionModel . . . . . . . . . . . . . . . . . . . . . . 267
12.2 Estimationand Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
12.3 ModelUtilityand Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
12.4 PolynomialRegression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
12.5 Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
12.6 QualitativeExplanatoryVariables . . . . . . . . . . . . . . . . . . . . . . . . . 286
12.7 PartialF Statistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
12.8 ResidualAnalysisand DiagnosticTools . . . . . . . . . . . . . . . . . . . . . 291
12.9 AdditionalTopics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
13 Resampling Methods 297
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
13.2 BootstrapStandard Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
13.3 BootstrapConfidence Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . 303
13.4 ResamplinginHypothesisTests . . . . . . . . . . . . . . . . . . . . . . . . . 305
Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
14 Categorical DataAnalysis 311
15 Nonparametric Statistics 313
16 Time Series 315
A R SessionInformation 317
B GNU FreeDocumentation License 319
C History 327
D Data 329
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D.1 DataStructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
D.2 ImportingData . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
D.3 Creating NewDataSets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
D.4 EditingData . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
D.5 ExportingData . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336
D.6 ReshapingData . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
E Mathematical Machinery 339
E.1 Set Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
E.2 DifferentialandIntegral Calculus . . . . . . . . . . . . . . . . . . . . . . . . . 340
E.3 Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
E.4 TheGammaFunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
E.5 LinearAlgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
E.6 MultivariableCalculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
F Writing Reports withR 349
F.1 Whatto Write . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
F.2 HowtoWriteIt withR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
F.3 FormattingTables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
F.4 OtherFormats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
G Instructions forInstructors 355
G.1 GeneratingThisDocument . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
G.2 HowtoUseThisDocument . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
G.3 AncillaryMaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
G.4 ModifyingThisDocument . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
H RcmdrTestDrive Story 359
Bibliography 363
Index 369
Preface
This book was expanded from lecture materials I use in a one semester upper-division under-
graduate course entitled Probabilityand Statistics at Youngstown State University. Those lec-
ture materials, in turn, were based on notes that I transcribed as a graduate student at Bowling
Green State University. The course for which the materials were written is 50-50 Probabil-
ity and Statistics, and the attendees include mathematics, engineering, and computer science
majors(amongothers). Thecatalogprerequisitesforthecoursearea fullyearofcalculus.
The book can be subdividedinto three basic parts. The first part includes the introductions
and elementary descriptive statistics; I want the students to be knee-deep in data right out of
the gate. The second part is the study of probability, which begins at the basics of sets and
the equally likely model, journeys past discrete/continuous random variables, and continues
through to multivariate distributions. The chapter on sampling distributions paves the way to
thethirdpart,whichisinferentialstatistics. Thislastpartincludespointandintervalestimation,
hypothesistesting,and finisheswithintroductionstoselected topicsin appliedstatistics.
I usually only have time in one semester to cover a small subset of this book. I cover the
material in Chapter 2 in a class period that is supplemented by a take-home assignment for
the students. I spend a lot of time on Data Description, Probability, Discrete, and Continuous
Distributions. I mention selected facts from Multivariate Distributions in passing, and discuss
the meaty parts of Sampling Distributions before moving right along to Estimation (which is
anotherchapterIdwellonconsiderably). HypothesisTestinggoesfasterafteralloftheprevious
work, and by that time the end of the semester is in sight. I normally choose one or two final
chapters (sometimes three) from the remaining to survey, and regret at the end that I did not
havethechance tocovermore.
InanattempttobecorrectIhaveincludedmaterialinthisbookwhichIwouldnormallynot
mention during the course of a standard lecture. For instance, I normally do not highlight the
intricaciesofmeasuretheoryorintegrabilityconditionswhenspeakingtotheclass. Moreover,I
oftenstrayfromthematrixapproachtomultiplelinearregressionbecausemanyofmystudents
have not yet been formally trained in linear algebra. That being said, it is important to me for
the students to hold something in their hands which acknowledges the world of mathematics
and statistics beyond the classroom, and which may be useful to them for many semesters to
come. Italso mirrorsmyownexperienceas astudent.
Thevisionforthisdocumentisamoreorlessselfcontained,essentiallycomplete,correct,
introductory textbook. There should be plenty of exercises for the student, with full solutions
for some, and no solutions for others (so that the instructor may assign them for grading).
By Sweave’s dynamic nature it is possible to write randomly generated exercises and I had
plannedto implementthisideaalready throughoutthebook. Alas, thereare only24hours ina
day. Lookformorein futureeditions.
Seasoned readers will be able to detect my origins: Probability and Statistical Inference
by Hogg and Tanis [44], Statistical Inference by Casella and Berger [13], and Theory of Point
Estimation/Testing Statistical Hypotheses by Lehmann [59, 58]. I highly recommend each of
vii
viii CONTENTS
those books to every reader of this one. Some R books with “introductory” in the title that I
recommend are Introductory Statistics withRby Dalgaard [19] and UsingRfor Introductory
Statistics by Verzani [87]. Surely there are many, many other good introductory books about
R, but frankly, I have tried to steer clear of them for the past year or so to avoid any undue
influenceon myownwriting.
Iwouldliketomakespecialmentionoftwootherbooks: IntroductiontoStatisticalThought
by Michael Lavine [56] and Introduction to Probability by Grinstead and Snell [37]. Both of
thesebooksarefreeandarewhatultimatelyconvincedmetoreleaseIPURunderafreelicense,
S
too.
Please bear in mind that the title of this book is “Introduction to Probability and Statistics
UsingR”, and not“IntroductiontoR UsingProbabilityand Statistics”,noreven“Introduction
to Probability and Statistics and R Using Words”. The people at the party are Probability
and Statistics; the handshake is R. There are several important topics about R which some
individualswillfeelareunderdeveloped,glossedover,orwantonlyomitted. Somewillfeelthe
samewayabouttheprobabilisticand/orstatisticalcontent. StillotherswilljustwanttolearnR
and skipallofthemathematics.
Despite any misgivings: here it is, warts and all. I humbly invite said individuals to take
thisbook,withtheGNUFreeDocumentationLicense(GNU-FDL)inhand,andmakeitbetter.
In thatspiritthereare atleast afew waysin myviewin whichthisbookcouldbeimproved.
Better data. The data analyzed in this book are almost entirely from the datasets package
inbaseR, and here iswhy:
1. I madeaconsciouseffort tominimizedependenceoncontributedpackages,
2. The data are instantly available, already in the correct format, so we need not take
timetomanagethem,and
3. Thedataare real.
I made no attempt to choose data sets that would be interesting to the students; rather,
data were chosen for their potential to convey a statistical point. Many of the data sets
aredecadesoldormore(forinstance,thedatausedtointroducesimplelinearregression
arethespeeds andstoppingdistances ofcars inthe1920’s).
InaperfectworldwithinfinitetimeIwouldresearchandcontributerecent,realdataina
contextcrafted toengagethestudentsin everyexample. Oneday Ihopeto stumbleover
saidtime. In themeantime,I willadd new datasetsincrementallyas timepermits.
Moreproofs. I would like to include more proofs for the sake of completeness (I understand
that some people would not consider more proofs to be improvement). Many proofs
have been skipped entirely, and I am not aware of any rhyme or reason to the current
omissions. Iwilladd morewhenIget achance.
Moreand better graphics: Ihavenotusedtheggplot2 package[90]becauseIdonotknow
howtouseityet. It isonmy to-dolist.
Moreand better exercises: There are only a few exercises in thefirst editionsimplybecause
Ihavenothadtimetowritemore. Ihavetoyedwiththeexamspackage[38]andIbelieve
that it is a right way to moveforward. As I learn more about what the package can do I
wouldliketo incorporateit intolatereditionsofthisbook.
CONTENTS ix
About This Document
IPURcontainsmanyinterrelatedparts: theDocument,theProgram,thePackage,andtheAn-
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> install.packages("IPSUR")
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Notation
We use the notation x or stem.leaf notation to denote objects, functions, etc.. The sequence
“Statistics ⊲Summaries⊲Active Dataset”meanstoclicktheStatistics menuitem,nextclick
theSummariessubmenuitem,and finallyclick Active Dataset.