The Probability Lifesaver: All the Tools You Need to Understand Chance

753 Pages·2017·3.74 MB·English

by  Steven J. Miller

The Probability Lifesaver
APRINCETONLIFESAVERSTUDYGUIDE
TheCalculusLifesaver:AlltheToolsYouNeedtoExcelatCalculusbyAdrianBanner
TheRealAnalysisLifesaver:AlltheToolsYouNeedtoUnderstandProofsbyRaffiGrinberg
TheProbabilityLifesaver:AlltheToolsYouNeedtoUnderstandChancebyStevenJ.Miller
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All the tools you need to understand chance
Steven J. Miller
PRINCETON UNIVERSITY PRESS
Princeton and Oxford
Copyright(cid:2)c 2017byPrincetonUniversityPress
PublishedbyPrincetonUniversityPress,41WilliamStreet,
Princeton,NewJersey08540
IntheUnitedKingdom:PrincetonUniversityPress,6OxfordStreet,
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press.princeton.edu
CoverillustrationbyDimitriKaretnikov;
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coverdesignbyLorraineBetzDoneker
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MATLABR isaregisteredtrademarkofTheMathWorksInc.andisusedwith
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thisbook.Thisbook’suseofaMATLABR relatedproductsdoesnotconstitute
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LibraryofCongressCataloging-in-PublicationData
Names:Miller,StevenJ.,1974–
Title:Theprobabilitylifesaver:allthetoolsyouneedtounderstand
chance/StevenJ.Miller.
Description:Princeton:PrincetonUniversityPress,[2017]|Series:A
Princetonlifesaverstudyguide|Includesbibliographicalreferencesandindex.
Identifiers:LCCN2016040785|ISBN9780691149547(hardcover:alk.paper)|
ISBN9780691149554(pbk.:alk.paper)
Subjects:LCSH:Probabilities.|Chance.|Gamesofchance(Mathematics)|
Randomvariables.
Classification:LCCQA273.M551852017|DDC519.2-dc23LCrecord
availableathttps://lccn.loc.gov/2016040785
BritishLibraryCataloging-in-PublicationDataisavailable
ThisbookhasbeencomposedinTimesNewRomanwithStencilandAvantGarde
Printedonacid-freepaper.∞
TypesetbyNovaTechsetPvtLtd,Bangalore,India
PrintedintheUnitedStatesofAmerica
13579108642
Contents
NotetoReaders xv
HowtoUseThisBook xix
I GeneralTheory 1
1 Introduction 3
1.1 BirthdayProblem 4
1.1.1 StatingtheProblem 4
1.1.2 SolvingtheProblem 6
1.1.3 GeneralizingtheProblemandSolution:Efficiencies 11
1.1.4 NumericalTest 14
1.2 FromShootingHoopstotheGeometricSeries 16
1.2.1 TheProblemandItsSolution 16
1.2.2 RelatedProblems 21
1.2.3 GeneralProblemSolvingTips 25
1.3 Gambling 27
1.3.1 The2008SuperBowlWager 28
1.3.2 ExpectedReturns 28
1.3.3 TheValueofHedging 29
1.3.4 Consequences 31
1.4 Summary 31
1.5 Exercises 34
2 BasicProbabilityLaws 40
2.1 Paradoxes 41
2.2 SetTheoryReview 43
2.2.1 CodingDigression 47
2.2.2 SizesofInfinityandProbabilities 48
2.2.3 OpenandClosedSets 50
2.3 OutcomeSpaces,Events,andtheAxiomsofProbability 52
2.4 AxiomsofProbability 57
2.5 BasicProbabilityRules 59
2.5.1 LawofTotalProbability 60
2.5.2 ProbabilitiesofUnions 61
2.5.3 ProbabilitiesofInclusions 64
2.6 ProbabilitySpacesandσ-algebras 65
vi • Contents
2.7 Appendix:ExperimentallyFindingFormulas 70
2.7.1 ProductRuleforDerivatives 71
2.7.2 ProbabilityofaUnion 72
2.8 Summary 73
2.9 Exercises 73
3 CountingI:Cards 78
3.1 FactorialsandBinomialCoefficients 79
3.1.1 TheFactorialFunction 79
3.1.2 BinomialCoefficients 82
3.1.3 Summary 87
3.2 Poker 88
3.2.1 Rules 88
3.2.2 Nothing 90
3.2.3 Pair 92
3.2.4 TwoPair 95
3.2.5 ThreeofaKind 96
3.2.6 Straights,Flushes,andStraightFlushes 96
3.2.7 FullHouseandFourofaKind 97
3.2.8 PracticePokerHand:I 98
3.2.9 PracticePokerHand:II 100
3.3 Solitaire 101
3.3.1 Klondike 102
3.3.2 AcesUp 105
3.3.3 FreeCell 107
3.4 Bridge 108
3.4.1 Tic-tac-toe 109
3.4.2 NumberofBridgeDeals 111
3.4.3 TrumpSplits 117
3.5 Appendix:CodingtoComputeProbabilities 120
3.5.1 TrumpSplitandCode 120
3.5.2 PokerHandCodes 121
3.6 Summary 124
3.7 Exercises 124
4 ConditionalProbability,Independence,and
Bayes’Theorem 128
4.1 ConditionalProbabilities 129
4.1.1 GuessingtheConditionalProbabilityFormula 131
4.1.2 ExpectedCountsApproach 132
4.1.3 VennDiagramApproach 133
4.1.4 TheMontyHallProblem 135
4.2 TheGeneralMultiplicationRule 136
4.2.1 Statement 136
4.2.2 PokerExample 136
4.2.3 HatProblemandErrorCorrectingCodes 138
4.2.4 AdvancedRemark:DefinitionofConditionalProbability 138
4.3 Independence 139
4.4 Bayes’Theorem 142
Contents • vii
4.5 PartitionsandtheLawofTotalProbability 147
4.6 Bayes’TheoremRevisited 150
4.7 Summary 151
4.8 Exercises 152
5 CountingII:Inclusion-Exclusion 156
5.1 FactorialandBinomialProblems 157
5.1.1 “Howmany”versus“What’stheprobability” 157
5.1.2 ChoosingGroups 159
5.1.3 CircularOrderings 160
5.1.4 ChoosingEnsembles 162
5.2 TheMethodofInclusion-Exclusion 163
5.2.1 SpecialCasesoftheInclusion-ExclusionPrinciple 164
5.2.2 StatementoftheInclusion-ExclusionPrinciple 167
5.2.3 JustificationoftheInclusion-ExclusionFormula 168
5.2.4 UsingInclusion-Exclusion:SuitedHand 171
5.2.5 TheAtLeasttoExactlyMethod 173
5.3 Derangements 176
5.3.1 CountingDerangements 176
5.3.2 TheProbabilityofaDerangement 178
5.3.3 CodingDerangementExperiments 178
5.3.4 ApplicationsofDerangements 179
5.4 Summary 181
5.5 Exercises 182
6 CountingIII:AdvancedCombinatorics 186
6.1 BasicCounting 187
6.1.1 EnumeratingCases:I 187
6.1.2 EnumeratingCases:II 188
6.1.3 SamplingWithandWithoutReplacement 192
6.2 WordOrderings 199
6.2.1 CountingOrderings 200
6.2.2 MultinomialCoefficients 202
6.3 Partitions 205
6.3.1 TheCookieProblem 205
6.3.2 Lotteries 207
6.3.3 AdditionalPartitions 212
6.4 Summary 214
6.5 Exercises 215
II IntroductiontoRandomVariables 219
7 IntroductiontoDiscreteRandomVariables 221
7.1 DiscreteRandomVariables:Definition 221
7.2 DiscreteRandomVariables:PDFs 223
7.3 DiscreteRandomVariables:CDFs 226
viii • Contents
7.4 Summary 233
7.5 Exercises 235
8 IntroductiontoContinuousRandomVariables 238
8.1 FundamentalTheoremofCalculus 239
8.2 PDFsandCDFs:Definitions 241
8.3 PDFsandCDFs:Examples 243
8.4 ProbabilitiesofSingletonEvents 248
8.5 Summary 250
8.6 Exercises 250
9 Tools:Expectation 254
9.1 CalculusMotivation 255
9.2 ExpectedValuesandMoments 257
9.3 MeanandVariance 261
9.4 JointDistributions 265
9.5 LinearityofExpectation 269
9.6 PropertiesoftheMeanandtheVariance 274
9.7 SkewnessandKurtosis 279
9.8 Covariances 280
9.9 Summary 281
9.10 Exercises 281
10 Tools:ConvolutionsandChangingVariables 285
10.1 Convolutions:DefinitionsandProperties 286
10.2 Convolutions:DieExample 289
10.2.1 TheoreticalCalculation 289
10.2.2 ConvolutionCode 290
10.3 ConvolutionsofSeveralVariables 291
10.4 ChangeofVariableFormula:Statement 294
10.5 ChangeofVariablesFormula:Proof 297
10.6 Appendix:ProductsandQuotients
ofRandomVariables 302
10.6.1 DensityofaProduct 302
10.6.2 DensityofaQuotient 303
10.6.3 Example:QuotientofExponentials 304
10.7 Summary 305
10.8 Exercises 305
11 Tools:DifferentiatingIdentities 309
11.1 GeometricSeriesExample 310
11.2 MethodofDifferentiatingIdentities 313
11.3 ApplicationstoBinomialRandomVariables 314
11.4 ApplicationstoNormalRandomVariables 317
Contents • ix
11.5 ApplicationstoExponential
RandomVariables 320
11.6 Summary 322
11.7 Exercises 323
III SpecialDistributions 325
12 DiscreteDistributions 327
12.1 TheBernoulliDistribution 328
12.2 TheBinomialDistribution 328
12.3 TheMultinomialDistribution 332
12.4 TheGeometricDistribution 335
12.5 TheNegativeBinomialDistribution 336
12.6 ThePoissonDistribution 340
12.7 TheDiscreteUniformDistribution 343
12.8 Exercises 346
13 ContinuousRandomVariables:
UniformandExponential 349
13.1 TheUniformDistribution 349
13.1.1 MeanandVariance 350
13.1.2 SumsofUniformRandomVariables 352
13.1.3 Examples 354
13.1.4 GeneratingRandomNumbersUniformly 356
13.2 TheExponentialDistribution 357
13.2.1 MeanandVariance 357
13.2.2 SumsofExponentialRandomVariables 361
13.2.3 ExamplesandApplicationsofExponentialRandom
Variables 364
13.2.4 GeneratingRandomNumbersfrom
ExponentialDistributions 365
13.3 Exercises 367
14 ContinuousRandomVariables:TheNormalDistribution 371
14.1 DeterminingtheNormalizationConstant 372
14.2 MeanandVariance 375
14.3 SumsofNormalRandomVariables 379
14.3.1 Case1:μ =μ =0andσ2 =σ2 =1 380
X Y X Y
14.3.2 Case2:Generalμ ,μ andσ2,σ2 383
X Y X Y
14.3.3 SumsofTwoNormals:FasterAlgebra 385
14.4 GeneratingRandomNumbersfrom
NormalDistributions 386
14.5 ExamplesandtheCentralLimitTheorem 392
14.6 Exercises 393