Mathematical Statistics With Applications.

849 Pages·2009·4.8 MB·English

Mathematical Statistics with
Applications
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Mathematical Statistics with Applications
by Kandethody M. Ramachandran and Chris P. Tsokos
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Mathematical Statistics with
Applications
KandethodyM.Ramachandran
DepartmentofMathematicsandStatistics
UniversityofSouthFlorida
Tampa,FL
ChrisP.Tsokos
DepartmentofMathematicsandStatistics
UniversityofSouthFlorida
Tampa,FL
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Ramachandran,K.M.
Mathematicalstatisticswithapplications/KandethodyM.Ramachandran,ChrisP.Tsokos.
p.cm.
ISBN978-0-12-374848-5(hardcover:alk.paper)
1.Mathematicalstatistics.2.Mathematical
statistics—Dataprocessing.I.Tsokos,ChrisP.II.Title.
QA276.R3282009
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2008044556
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Dedicated to our families:
Usha, Vikas, Vilas, and Varsha Ramachandran
and
Debbie, Matthew, Jonathan, and Maria Tsokos
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Contents
Preface................................................................................................... xv
Acknowledgments ...................................................................................... xix
AbouttheAuthors....................................................................................... xxi
FlowChart ..............................................................................................xxiii
CHAPTER 1 DescriptiveStatistics............................................................. 1
1.1 Introduction...................................................................... 2
1.1.1 DataCollection......................................................... 3
1.2 BasicConcepts .................................................................. 3
1.2.1 TypesofData........................................................... 5
1.3 SamplingSchemes .............................................................. 8
1.3.1 ErrorsinSampleData.................................................. 11
1.3.2 SampleSize............................................................. 12
1.4 GraphicalRepresentationofData .............................................. 13
1.5 NumericalDescriptionofData ................................................. 26
1.5.1 NumericalMeasuresforGroupedData............................... 30
1.5.2 BoxPlots ............................................................... 33
1.6 ComputersandStatistics........................................................ 39
1.7 ChapterSummary ............................................................... 40
1.8 ComputerExamples............................................................. 41
1.8.1 MinitabExamples...................................................... 41
1.8.2 SPSSExamples ........................................................ 46
1.8.3 SASExamples.......................................................... 47
ProjectsforChapter1................................................................ 51
CHAPTER 2 BasicConceptsfromProbabilityTheory .......................................... 53
2.1 Introduction...................................................................... 54
2.2 RandomEventsandProbability ................................................ 55
2.3 CountingTechniquesandCalculationofProbabilities........................ 63
2.4 TheConditionalProbability,Independence,andBayes’Rule ................ 71
2.5 RandomVariablesandProbabilityDistributions .............................. 83
2.6 MomentsandMoment-GeneratingFunctions ................................. 92
2.6.1 SkewnessandKurtosis................................................. 98
2.7 ChapterSummary ............................................................... 107
2.8 ComputerExamples(Optional)................................................. 108
2.8.1 MinitabComputations ................................................. 109
2.8.2 SPSSExamples ........................................................ 110
2.8.3 SASExamples.......................................................... 110
ProjectsforChapter2................................................................ 112
vii
viii Contents
CHAPTER 3 AdditionalTopicsinProbability .................................................. 113
3.1 Introduction...................................................................... 114
3.2 SpecialDistributionFunctions.................................................. 114
3.2.1 TheBinomialProbabilityDistribution................................ 114
3.2.2 PoissonProbabilityDistribution....................................... 119
3.2.3 UniformProbabilityDistribution...................................... 122
3.2.4 NormalProbabilityDistribution....................................... 125
3.2.5 GammaProbabilityDistribution ...................................... 131
3.3 JointProbabilityDistributions.................................................. 141
3.3.1 CovarianceandCorrelation............................................ 148
3.4 FunctionsofRandomVariables................................................. 154
3.4.1 MethodofDistributionFunctions..................................... 154
3.4.2 ThepdfofY =g(X),WheregIsDifferentiableandMonotone
IncreasingorDecreasing............................................... 156
3.4.3 ProbabilityIntegralTransformation................................... 157
3.4.4 FunctionsofSeveralRandomVariables:MethodofDistribution
Functions ............................................................... 158
3.4.5 TransformationMethod................................................ 159
3.5 LimitTheorems.................................................................. 163
3.6 ChapterSummary ............................................................... 173
3.7 ComputerExamples(Optional)................................................. 175
3.7.1 MinitabExamples...................................................... 175
3.7.2 SPSSExamples ........................................................ 177
3.7.3 SASExamples.......................................................... 178
ProjectsforChapter3................................................................ 180
CHAPTER 4 SamplingDistributions ..........................................................183
4.1 Introduction...................................................................... 184
4.1.1 FinitePopulation....................................................... 187
4.2 SamplingDistributionsAssociatedwithNormalPopulations................. 191
4.2.1 Chi-SquareDistribution................................................ 192
4.2.2 Studentt-Distribution.................................................. 198
4.2.3 F-Distribution .......................................................... 202
4.3 OrderStatistics .................................................................. 207
4.4 LargeSampleApproximations.................................................. 212
4.4.1 TheNormalApproximationtotheBinomialDistribution ........... 213
4.5 ChapterSummary ............................................................... 218
4.6 ComputerExamples............................................................. 219
4.6.1 MinitabExamples...................................................... 219
4.6.2 SPSSExamples ........................................................ 219
4.6.3 SASExamples.......................................................... 219
ProjectsforChapter4................................................................ 221
Contents ix
CHAPTER 5 PointEstimation.................................................................225
5.1 Introduction...................................................................... 226
5.2 TheMethodofMoments........................................................ 227
5.3 TheMethodofMaximumLikelihood.......................................... 235
5.4 SomeDesirablePropertiesofPointEstimators................................ 246
5.4.1 UnbiasedEstimators ................................................... 247
5.4.2 Sufficiency.............................................................. 252
5.5 OtherDesirablePropertiesofaPointEstimator............................... 266
5.5.1 Consistency............................................................. 266
5.5.2 Efficiency............................................................... 270
5.5.3 MinimalSufficiencyandMinimum-VarianceUnbiased
Estimation .............................................................. 277
5.6 ChapterSummary ............................................................... 282
5.7 ComputerExamples............................................................. 283
ProjectsforChapter5................................................................ 285
CHAPTER 6 IntervalEstimation .............................................................. 291
6.1 Introduction...................................................................... 292
6.1.1 AMethodofFindingtheConfidenceInterval:PivotalMethod...... 293
6.2 LargeSampleConfidenceIntervals:OneSampleCase ....................... 300
6.2.1 ConfidenceIntervalforProportion,p ................................. 302
6.2.2 MarginofErrorandSampleSize ..................................... 303
6.3 SmallSampleConfidenceIntervalsforμ...................................... 310
6.4 AConfidenceIntervalforthePopulationVariance ............................ 315
6.5 ConfidenceIntervalConcerningTwoPopulationParameters................. 321
6.6 ChapterSummary ............................................................... 330
6.7 ComputerExamples............................................................. 330
6.7.1 MinitabExamples...................................................... 330
6.7.2 SPSSExamples ........................................................ 332
6.7.3 SASExamples.......................................................... 333
ProjectsforChapter6................................................................ 334
CHAPTER 7 HypothesisTesting...............................................................337
7.1 Introduction...................................................................... 338
7.1.1 SampleSize............................................................. 346
7.2 TheNeyman–PearsonLemma.................................................. 349
7.3 LikelihoodRatioTests .......................................................... 355
7.4 HypothesesforaSingleParameter............................................. 361
7.4.1 Thep-Value............................................................. 361
7.4.2 HypothesisTestingforaSingleParameter............................ 363