Examples and Problems in Mathematical Statistics

654 Pages·2014·2.882 MB·English

by  Shelemyahu Zacks

Examples and Problems
in Mathematical Statistics
Shelemyahu Zacks

Examples and Problems
in Mathematical Statistics
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Examples and Problems
in Mathematical Statistics
SHELEMYAHUZACKS
DepartmentofMathematicalSciences
BinghamtonUniversity
Binghamton,NY
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Summary:“Thisbookpresentsexamplesthatillustratethetheoryofmathematicalstatisticsanddetails
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QC32.Z2652013
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2013034492
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Contents
Preface xv
ListofRandomVariables xvii
ListofAbbreviations xix
1 BasicProbabilityTheory 1
PARTI:THEORY, 1
1.1 OperationsonSets, 1
1.2 Algebraandσ-Fields, 2
1.3 ProbabilitySpaces, 4
1.4 ConditionalProbabilitiesandIndependence, 6
1.5 RandomVariablesandTheirDistributions, 8
1.6 TheLebesgueandStieltjesIntegrals, 12
1.6.1 GeneralDefinitionofExpectedValue:TheLebesgueIntegral, 12
1.6.2 TheStieltjes–RiemannIntegral, 17
1.6.3 MixturesofDiscreteandAbsolutelyContinuousDistributions, 19
1.6.4 QuantilesofDistributions, 19
1.6.5 Transformations, 20
1.7 JointDistributions,ConditionalDistributionsandIndependence, 21
1.7.1 JointDistributions, 21
1.7.2 ConditionalExpectations:GeneralDefinition, 23
1.7.3 Independence, 26
1.8 MomentsandRelatedFunctionals, 26
1.9 ModesofConvergence, 35
1.10 WeakConvergence, 39
1.11 LawsofLargeNumbers, 41
vii
viii CONTENTS
1.11.1 TheWeakLawofLargeNumbers(WLLN), 41
1.11.2 TheStrongLawofLargeNumbers(SLLN), 42
1.12 CentralLimitTheorem, 44
1.13 MiscellaneousResults, 47
1.13.1 LawoftheIteratedLogarithm, 48
1.13.2 UniformIntegrability, 48
1.13.3 Inequalities, 52
1.13.4 TheDeltaMethod, 53
1.13.5 TheSymbolso and O , 55
p p
1.13.6 TheEmpiricalDistributionandSampleQuantiles, 55
PARTII:EXAMPLES, 56
PARTIII:PROBLEMS, 73
PARTIV:SOLUTIONSTOSELECTEDPROBLEMS, 93
2 StatisticalDistributions 106
PARTI:THEORY, 106
2.1 IntroductoryRemarks, 106
2.2 FamiliesofDiscreteDistributions, 106
2.2.1 BinomialDistributions, 106
2.2.2 HypergeometricDistributions, 107
2.2.3 PoissonDistributions, 108
2.2.4 Geometric,Pascal,andNegativeBinomialDistributions, 108
2.3 SomeFamiliesofContinuousDistributions, 109
2.3.1 RectangularDistributions, 109
2.3.2 BetaDistributions, 111
2.3.3 GammaDistributions, 111
2.3.4 WeibullandExtremeValueDistributions, 112
2.3.5 NormalDistributions, 113
2.3.6 NormalApproximations, 114
2.4 Transformations, 118
2.4.1 One-to-OneTransformationsofSeveralVariables, 118
2.4.2 DistributionofSums, 118
2.4.3 DistributionofRatios, 118
2.5 VariancesandCovariancesofSampleMoments, 120
2.6 DiscreteMultivariateDistributions, 122
2.6.1 TheMultinomialDistribution, 122
2.6.2 MultivariateNegativeBinomial, 123
2.6.3 MultivariateHypergeometricDistributions, 124