INSTRUCTORS SOLUTION MANUAL FOR Probability and Statistics 4th Edition by Morris DeGroot , Mark Schervish ISBN:978-0134995472 ALL CHAPTERS COVERED YOUR ULTIMATE GUIDE 100% VERIFIED A+ GRADE ASSURED!!!!! NEW LATEST UPDATE!!!!!!

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MARK SCHERVISHVC




Carnegie Mellon University
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P ROBABILITY AND STATISTICS
V C V C




FOURTH E DITION VC




Morris DeGroot VC



Carnegie Mellon University
VC VC




Mark Schervish
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Carnegie Mellon University
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ngVCoutVCof,VCtheVCfurnishing,VCperformance,VCorVCuseVCofVCtheseVCprograms.

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WesleyVCfromVCelectronicVCfilesVCsuppliedVCbyVCtheVCauthor.VCCopyrightVC©VC2012,VC20

02,VC1986VCPearsonVCEducation,VCInc.
PublishingVCasVCPearsonVCAddison-Wesley,VC75VCArlingtonVCStreet,VCBoston,VCMAVC02116.

AllVCrightsVCreserved.VCThisVCmanualVCmayVCbeVCreproducedVCforVCclassroo

mVCuseVConly.VCISBN-13:VC978-0-321-71597-5
ISBN-10:V C 0-321-71597-7

,Contents

Preface .................................................................................................................................................................... vi

1 IntroductionVCtoVCProbability 1
1.2 InterpretationsV C ofV C ProbabilityV C V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .
V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C . 1
1.4 SetV C TheoryV C V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V
C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C . 1
1.5 TheV C DefinitionV C ofV C ProbabilityV C V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V
C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C . 3
1.6 FiniteV C SampleV C SpacesV C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V
C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C . 6
1.7 CountingV C MethodsV C V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V
C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C . 7
1.8 CombinatorialV C MethodsV C V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C
.V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C .V C . 8
1.9 MultinomialV C Coefficients ...........................................................................................................................13
1.10 TheVCProbabilityVCofVCaVCUnionVCofVCEvents .............................................................................................. 16
1.12V C SupplementaryV C Exercises .........................................................................................................................20

2 ConditionalV C Probability 25
2.2 TheV C DefinitionV C ofV C ConditionalV C Probability .........................................................................................25
2.3 IndependentV C Events ....................................................................................................................................28
2.4 Bayes’V C Theorem ..........................................................................................................................................34
2.5 TheV C Gambler’sV C RuinV C Problem ..............................................................................................................40
2.6 SupplementaryV C Exercises ...........................................................................................................................41

3 RandomV C VariablesV C andV C Distributions 49
3.2 RandomVCVariablesVCandVCDiscreteVCDistributions.................................................................................... 49
3.3 ContinuousV C Distributions ............................................................................................................................50
3.4 TheVCCumulativeVCDistributionVCFunction ................................................................................................ 53
3.5 BivariateVCDistributions .............................................................................................................................. 58
3.6 MarginalVCDistributions ............................................................................................................................... 64
3.7 ConditionalV C Distributions ......................................................................................................................... 70
3.8 MultivariateV C Distributions........................................................................................................................ 76
3.9 FunctionsV C ofV C aV C RandomV C Variable .........................................................................................................81
3.10 FunctionsV C ofV C TwoV C orV C MoreV C RandomV C Variables ...............................................................................85
3.11 MarkovVCChains ........................................................................................................................................... 93
3.12 SupplementaryV C Exercises ...........................................................................................................................97

4 Expectation 107
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n-Wesley.

, 4.2 TheVCExpectationVCofVCaVCRandomVCVariable .......................................................................................... 107
4.3 PropertiesV C ofV C Expectations ...................................................................................................................... 110
4.4 Variance ....................................................................................................................................................... 113
4.5 Moments...................................................................................................................................................... 115
4.6 TheV C MeanV C andV C theV C Median................................................................................................................. 118

4.7 CovarianceV C andV C Correlation.................................................................................................................... 121
4.8 ConditionalV C Expectation ......................................................................................................................... 124
4.9 Utility.......................................................................................................................................................... 129
4.10 SupplementaryV C Exercises ......................................................................................................................... 134

5 SpecialV C Distributions 141
5.2 TheVCBernoulliVCandVCBinomialVCDistributions ....................................................................................... 141
5.3 TheV C HypergeometricV C Distributions ........................................................................................................ 145
5.4 TheVCPoissonVCDistributions ..................................................................................................................... 149
5.5 TheVCNegativeVCBinomialVCDistributions .................................................................................................. 155
5.6 TheVCNormalVCDistributions ..................................................................................................................... 159
5.7 TheVC GammaVC Distributions .................................................................................................................... 165
5.8 TheV C BetaV C Distributions .......................................................................................................................... 171
5.9 TheVCMultinomialVCDistributions ............................................................................................................. 174
5.10 TheVCBivariateVC NormalVC Distributions .................................................................................................... 177
5.11 SupplementaryV C Exercises ......................................................................................................................... 182

6 LargeV C RandomV C Samples 187
6.2 Introduction ................................................................................................................................................. 187
6.3 TheV C LawV C ofV C LargeV C Numbers .............................................................................................................. 188
6.4 TheV C CentralV C LimitV C Theorem .............................................................................................................. 194
6.5 TheV C CorrectionV C forV C Continuity ............................................................................................................. 198
6.6 SupplementaryV C Exercises ......................................................................................................................... 199

7 Estimation 203
7.2 StatisticalV C Inference ................................................................................................................................. 203
7.3 PriorVCandVCPosteriorVCDistributions ......................................................................................................... 204
7.4 ConjugateVCPriorVCDistributions .............................................................................................................. 207
7.5 BayesV C Estimators ....................................................................................................................................... 214
7.6 MaximumV C LikelihoodV C Estimators ......................................................................................................... 217
7.7 PropertiesV C ofV C MaximumV C LikelihoodV C Estimators .............................................................................. 220
7.8 SufficientV C Statistics .................................................................................................................................... 225
7.9 JointlyVCSufficientVCStatistics ..................................................................................................................... 228
7.10 ImprovingVCanVCEstimator ......................................................................................................................... 230
7.11 SupplementaryV C Exercises ......................................................................................................................... 234

8 SamplingV C DistributionsV C ofV C Estimators 239
8.2 TheVCSamplingVCDistributionVCofVCaVCStatistic ......................................................................................... 239
8.3 TheV C Chi-SquareV C Distributions................................................................................................................ 241
8.4 JointVCDistributionVCofVCtheVCSampleVCMeanVCandVCSampleVCVariance .................................................. 245
8.5 TheV C tV C Distributions ................................................................................................................................ 247
8.6 ConfidenceV C Intervals ................................................................................................................................. 250
8.7 BayesianVCAnalysis VC ofVCSamplesVCfromVCaVCNormalVCDistribution......................................................... 254
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n-Wesley.