Mathematical Statistics with Applications

939 Pages·2010·5.1 MB·English

by  Dennis Wackerly

Continuous Distributions
Moment-
Generating
Distribution ProbabilityFunction Mean Variance Function
Uniform f(y)= 1 ;θ ≤y≤θ θ1+θ2 (θ2−θ1)2 etθ2−etθ1
θ −θ 1 2 2 12 t(θ −θ )
2 1 2 1
(cid:1) (cid:2) (cid:3) (cid:4) (cid:5)
1 1 t2σ2
Normal f(y)= √ exp − (y−µ)2 µ σ2 exp µt+
σ 2π 2σ2 2
−∞<y<+∞
Exponential f(y)= 1e−y/β; β>0 β β2 (1−βt)−1
β
0<y<∞
(cid:1) (cid:4)
Gamma f(y)= (cid:5)(α1)βα yα−1e−y/β; αβ αβ2 (1−βt)−α
0<y<∞
Chi-square f(y)= (y)(v/2)−1e−y/2; v 2v (1−2t)−v/2
2v/2(cid:5)(v/2)
y2>0
(cid:1) (cid:4)
(cid:5)(α+β) α αβ
Beta f(y)= yα−1(1−y)β−1; doesnotexistin
(cid:5)(α)(cid:5)(β) α+β (α+β)2(α+β+1)
closedform
0<y<1
Discrete Distributions
Moment-
Generating
Distribution ProbabilityFunction Mean Variance Function
(cid:6) (cid:7)
Binomial p(y)= n py(1−p)n−y; np np(1−p) [pet+(1−p)]n
y
y=0,1,...,n
Geometric p(y)= p(1−p)y−1; 1 1−p pet
p p2 1−(1−p)et
y=1,2,...
(cid:6) (cid:7)(cid:6) (cid:7)
r N−r (cid:6) (cid:7)(cid:2) (cid:3)(cid:2) (cid:3)
Hypergeometric p(y)= y(cid:6) n(cid:7)−y ; nr n r N−r N−n
N N N N N−1
n
y=0,1,...,nifn≤r,
y=0,1,...,rifn>r
λye−λ
Poisson p(y)= ; λ λ exp[λ(et−1)]
y!
y=0,1,2,...
Negativebinomial p(y)=(cid:6)y−1(cid:7)pr(1−p)y−r; r r(1−p) (cid:1) pet (cid:4)r
r−1 p p2 1−(1−p)et
y=r,r+1,...
MATHEMATICAL STATISTICS WITH APPLICATIONS
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SEVENTH EDITION
Mathematical
Statistics with
Applications
Dennis D. Wackerly
UniversityofFlorida
William Mendenhall III
UniversityofFlorida,Emeritus
Richard L. Scheaffer
UniversityofFlorida,Emeritus
Australia•Brazil•Canada•Mexico•Singapore•Spain
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MathematicalStatisticswithApplications,SeventhEdition
DennisD.Wackerly,WilliamMendenhallIII,RichardL.Scheaffer
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InternationalStudentEdition
ISBN-13:978-0-495-38508-0
ISBN-10:0-495-38508-5
CONTENTS
Preface xiii
NotetotheStudent xxi
1 What Is Statistics? 1
1.1 Introduction 1
1.2 CharacterizingaSetofMeasurements:GraphicalMethods 3
1.3 CharacterizingaSetofMeasurements:NumericalMethods 8
1.4 HowInferencesAreMade 13
1.5 TheoryandReality 14
1.6 Summary 15
2 Probability 20
2.1 Introduction 20
2.2 ProbabilityandInference 21
2.3 AReviewofSetNotation 23
2.4 AProbabilisticModelforanExperiment:TheDiscreteCase 26
2.5 CalculatingtheProbabilityofanEvent:TheSample-PointMethod 35
2.6 ToolsforCountingSamplePoints 40
2.7 ConditionalProbabilityandtheIndependenceofEvents 51
2.8 TwoLawsofProbability 57
v
vi Contents
2.9 CalculatingtheProbabilityofanEvent:TheEvent-Composition
Method 62
2.10 TheLawofTotalProbabilityandBayes’Rule 70
2.11 NumericalEventsandRandomVariables 75
2.12 RandomSampling 77
2.13 Summary 79
3 Discrete Random Variables and Their
Probability Distributions 86
3.1 BasicDefinition 86
3.2 TheProbabilityDistributionforaDiscreteRandomVariable 87
3.3 TheExpectedValueofaRandomVariableoraFunction
ofaRandomVariable 91
3.4 TheBinomialProbabilityDistribution 100
3.5 TheGeometricProbabilityDistribution 114
3.6 TheNegativeBinomialProbabilityDistribution(Optional) 121
3.7 TheHypergeometricProbabilityDistribution 125
3.8 ThePoissonProbabilityDistribution 131
3.9 MomentsandMoment-GeneratingFunctions 138
3.10 Probability-GeneratingFunctions(Optional) 143
3.11 Tchebysheff’sTheorem 146
3.12 Summary 149
4 Continuous Variables and Their Probability
Distributions 157
4.1 Introduction 157
4.2 TheProbabilityDistributionforaContinuousRandomVariable 158
4.3 ExpectedValuesforContinuousRandomVariables 170
4.4 TheUniformProbabilityDistribution 174
4.5 TheNormalProbabilityDistribution 178
4.6 TheGammaProbabilityDistribution 185
4.7 TheBetaProbabilityDistribution 194
Contents vii
4.8 SomeGeneralComments 201
4.9 OtherExpectedValues 202
4.10 Tchebysheff’sTheorem 207
4.11 ExpectationsofDiscontinuousFunctionsandMixedProbability
Distributions(Optional) 210
4.12 Summary 214
5 Multivariate Probability Distributions 223
5.1 Introduction 223
5.2 BivariateandMultivariateProbabilityDistributions 224
5.3 MarginalandConditionalProbabilityDistributions 235
5.4 IndependentRandomVariables 247
5.5 TheExpectedValueofaFunctionofRandomVariables 255
5.6 SpecialTheorems 258
5.7 TheCovarianceofTwoRandomVariables 264
5.8 TheExpectedValueandVarianceofLinearFunctions
ofRandomVariables 270
5.9 TheMultinomialProbabilityDistribution 279
5.10 TheBivariateNormalDistribution(Optional) 283
5.11 ConditionalExpectations 285
5.12 Summary 290
6 Functions of Random Variables 296
6.1 Introduction 296
6.2 FindingtheProbabilityDistributionofaFunction
ofRandomVariables 297
6.3 TheMethodofDistributionFunctions 298
6.4 TheMethodofTransformations 310
6.5 TheMethodofMoment-GeneratingFunctions 318
6.6 MultivariableTransformationsUsingJacobians(Optional) 325
6.7 OrderStatistics 333
6.8 Summary 341