An Introduction to Probability and Statistics

707 Pages·2015·11.55 MB·English

by  Vijay K. Rohatgi

AN INTRODUCTION TO PROBABILITY
AND STATISTICS
WILEYSERIESINPROBABILITYANDSTATISTICS
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AN INTRODUCTION TO
PROBABILITY AND
STATISTICS
Third Edition
VIJAYK.ROHATGI
A.K.Md.EHSANESSALEH
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Rohatgi,V.K.,1939-
Anintroductiontoprobabilitytheoryandmathematicalstatistics/VijayK.RohatgiandA.K.Md.Ehsanes
Saleh.–3rdedition.
pages cm
Includesindex.
ISBN978-1-118-79964-2(cloth)
1. Probabilities. 2. Mathematicalstatistics. I. Saleh,A.K.Md.Ehsanes. II. Title.
QA273.R562015
519.5–dc23
2015004848
Setin10/12ptsTimesLtStdbySPiGlobal,Pondicherry,India
PrintedintheUnitedStatesofAmerica
10 9 8 7 6 5 4 3 2 1
3 2015
ToBinaandShahidara.
CONTENTS
PREFACETOTHETHIRDEDITION xiii
PREFACETOTHESECONDEDITION xv
PREFACETOTHEFIRSTEDITION xvii
ACKNOWLEDGMENTS xix
ENUMERATIONOFTHEOREMSANDREFERENCES xxi
1 Probability 1
1.1 Introduction, 1
1.2 SampleSpace, 2
1.3 ProbabilityAxioms, 7
1.4 Combinatorics:ProbabilityonFiniteSampleSpaces, 20
1.5 ConditionalProbabilityandBayesTheorem, 26
1.6 IndependenceofEvents, 31
2 RandomVariablesandTheirProbabilityDistributions 39
2.1 Introduction, 39
2.2 RandomVariables, 39
2.3 ProbabilityDistributionofaRandomVariable, 42
2.4 DiscreteandContinuousRandomVariables, 47
2.5 FunctionsofaRandomVariable, 55
viii CONTENTS
3 MomentsandGeneratingFunctions 67
3.1 Introduction, 67
3.2 MomentsofaDistributionFunction, 67
3.3 GeneratingFunctions, 83
3.4 SomeMomentInequalities, 93
4 MultipleRandomVariables 99
4.1 Introduction, 99
4.2 MultipleRandomVariables, 99
4.3 IndependentRandomVariables, 114
4.4 FunctionsofSeveralRandomVariables, 123
4.5 Covariance,CorrelationandMoments, 143
4.6 ConditionalExpectation, 157
4.7 OrderStatisticsandTheirDistributions, 164
5 SomeSpecialDistributions 173
5.1 Introduction, 173
5.2 SomeDiscreteDistributions, 173
5.2.1 DegenerateDistribution, 173
5.2.2 Two-PointDistribution, 174
5.2.3 UniformDistributiononnPoints, 175
5.2.4 BinomialDistribution, 176
5.2.5 NegativeBinomialDistribution(PascalorWaitingTime
Distribution), 178
5.2.6 HypergeometricDistribution, 183
5.2.7 NegativeHypergeometricDistribution, 185
5.2.8 PoissonDistribution, 186
5.2.9 MultinomialDistribution, 189
5.2.10 MultivariateHypergeometricDistribution, 192
5.2.11 MultivariateNegativeBinomialDistribution, 192
5.3 SomeContinuousDistributions, 196
5.3.1 UniformDistribution(RectangularDistribution), 199
5.3.2 GammaDistribution, 202
5.3.3 BetaDistribution, 210
5.3.4 CauchyDistribution, 213
5.3.5 NormalDistribution(theGaussianLaw), 216
5.3.6 SomeOtherContinuousDistributions, 222
5.4 BivariateandMultivariateNormalDistributions, 228
5.5 ExponentialFamilyofDistributions, 240
6 SampleStatisticsandTheirDistributions 245
6.1 Introduction, 245
6.2 RandomSampling, 246
6.3 SampleCharacteristicsandTheirDistributions, 249
CONTENTS ix
6.4 Chi-Square,t-,andF-Distributions:ExactSamplingDistributions, 262
6.5 Distributionof(X,S2)inSamplingfromaNormalPopulation, 271
6.6 SamplingfromaBivariateNormalDistribution, 276
7 BasicAsymptotics:LargeSampleTheory 285
7.1 Introduction, 285
7.2 ModesofConvergence, 285
7.3 WeakLawofLargeNumbers, 302
7.4 StrongLawofLargeNumbers, 308
7.5 LimitingMomentGeneratingFunctions, 316
7.6 CentralLimitTheorem, 321
7.7 LargeSampleTheory, 331
8 ParametricPointEstimation 337
8.1 Introduction, 337
8.2 ProblemofPointEstimation, 338
8.3 Sufficiency,CompletenessandAncillarity, 342
8.4 UnbiasedEstimation, 359
8.5 UnbiasedEstimation(Continued):ALowerBoundfortheVariance
ofAnEstimator, 372
8.6 SubstitutionPrinciple(MethodofMoments), 386
8.7 MaximumLikelihoodEstimators, 388
8.8 BayesandMinimaxEstimation, 401
8.9 PrincipleofEquivariance, 418
9 Neyman–PearsonTheoryofTestingofHypotheses 429
9.1 Introduction, 429
9.2 SomeFundamentalNotionsofHypothesesTesting, 429
9.3 Neyman–PearsonLemma, 438
9.4 FamilieswithMonotoneLikelihoodRatio, 446
9.5 UnbiasedandInvariantTests, 453
9.6 LocallyMostPowerfulTests, 459
10 SomeFurtherResultsonHypothesesTesting 463
10.1 Introduction, 463
10.2 GeneralizedLikelihoodRatioTests, 463
10.3 Chi-SquareTests, 472
10.4 t-Tests, 484
10.5 F-Tests, 489
10.6 BayesandMinimaxProcedures, 491
x CONTENTS
11 ConfidenceEstimation 499
11.1 Introduction, 499
11.2 SomeFundamentalNotionsofConfidenceEstimation, 499
11.3 MethodsofFindingConfidenceIntervals, 504
11.4 Shortest-LengthConfidenceIntervals, 517
11.5 UnbiasedandEquivariantConfidenceIntervals, 523
11.6 Resampling:BootstrapMethod, 530
12 GeneralLinearHypothesis 535
12.1 Introduction, 535
12.2 GeneralLinearHypothesis, 535
12.3 RegressionAnalysis, 543
12.3.1 MultipleLinearRegression, 543
12.3.2 LogisticandPoissonRegression, 551
12.4 One-WayAnalysisofVariance, 554
12.5 Two-WayAnalysisofVariancewithOneObservationPerCell, 560
12.6 Two-WayAnalysisofVariancewithInteraction, 566
13 NonparametricStatisticalInference 575
13.1 Introduction, 575
13.2 U-Statistics, 576
13.3 SomeSingle-SampleProblems, 584
13.3.1 Goodness-of-FitProblem, 584
13.3.2 ProblemofLocation, 590
13.4 SomeTwo-SampleProblems, 599
13.4.1 MedianTest, 601
13.4.2 Kolmogorov–SmirnovTest, 602
13.4.3 TheMann–Whitney–WilcoxonTest, 604
13.5 TestsofIndependence, 608
13.5.1 Chi-squareTestofIndependence—ContingencyTables, 608
13.5.2 Kendall’sTau, 611
13.5.3 Spearman’sRankCorrelationCoefficient, 614
13.6 SomeApplicationsofOrderStatistics, 619
13.7 Robustness, 625
13.7.1 EffectofDeviationsfromModelAssumptionsonSome
ParametricProcedures, 625
13.7.2 SomeRobustProcedures, 631
FREQUENTLYUSEDSYMBOLSANDABBREVIATIONS 637
REFERENCES 641
CONTENTS xi
STATISTICALTABLES 647
ANSWERSTOSELECTEDPROBLEMS 667
AUTHORINDEX 677
SUBJECTINDEX 679