Fundamentals of Applied Probability and Random Processes, Second Edition

457 Pages·2014·3.15 MB·English

by  Oliver Ibe

Fundamentals of Applied Probability
and Random Processes
This page intentionally left blank
Fundamentals of Applied
Probability and Random
Processes
2nd Edition
Oliver C. Ibe
University ofMassachusetts,Lowell, Massachusetts
AMSTERDAM (cid:129) BOSTON (cid:129) HEIDELBERG (cid:129) LONDON
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Ibe,OliverC.(OliverChukwudi),1947-
Fundamentalsofappliedprobabilityandrandomprocesses/OliverIbe.–Secondedition.
pagescm
Includesbibliographicalreferencesandindex.
ISBN978-0-12-800852-2(alk.paper)
1.Probabilities.I.Title.
QA273.I242014
519.2–dc23
2014005103
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ISBN:978-0-12-800852-2
Contents
ACKNOWLEDGMENT................................................................................xiv
PREFACE TO THE SECOND EDITION........................................................xvi
PREFACE TO FIRST EDITION...................................................................xix
CHAPTER1 Basic Probability Concepts...................................................1
1.1 Introduction..............................................................................1
1.2 SampleSpaceandEvents.......................................................2
1.3 DefinitionsofProbability.........................................................4
1.3.1 AxiomaticDefinition.....................................................4
1.3.2 Relative-FrequencyDefinition.....................................4
1.3.3 ClassicalDefinition......................................................4
1.4 ApplicationsofProbability.......................................................6
1.4.1 InformationTheory.......................................................6
1.4.2 ReliabilityEngineering.................................................7
1.4.3 QualityControl.............................................................7
1.4.4 ChannelNoise..............................................................8
1.4.5 SystemSimulation.......................................................8
1.5 ElementarySetTheory............................................................9
1.5.1 SetOperations..............................................................9
1.5.2 NumberofSubsetsofaSet......................................10
1.5.3 VennDiagram.............................................................10
1.5.4 SetIdentities..............................................................11
1.5.5 DualityPrinciple.........................................................13
1.6 PropertiesofProbability........................................................13
1.7 ConditionalProbability...........................................................14
1.7.1 TotalProbabilityandtheBayes’Theorem................16
1.7.2 TreeDiagram.............................................................22
1.8 IndependentEvents...............................................................26
1.9 CombinedExperiments..........................................................29
v
vi Contents
1.10 BasicCombinatorialAnalysis..............................................30
1.10.1 Permutations..........................................................30
1.10.2 CircularArrangement............................................32
1.10.3 ApplicationsofPermutationsinProbability..........33
1.10.4 Combinations..........................................................34
1.10.5 TheBinomialTheorem...........................................37
1.10.6 Stirling’sFormula..................................................37
1.10.7 TheFundamentalCountingRule...........................38
1.10.8 ApplicationsofCombinationsinProbability..........40
1.11 ReliabilityApplications.........................................................41
1.12 ChapterSummary................................................................46
1.13 Problems..............................................................................46
Section1.2 SampleSpaceandEvents.............................46
Section1.3 DefinitionsofProbability...............................47
Section1.5 ElementarySetTheory..................................48
Section1.6 PropertiesofProbability................................50
Section1.7 ConditionalProbability...................................50
Section1.8 IndependentEvents.......................................52
Section1.10 CombinatorialAnalysis................................52
Section1.11 ReliabilityApplications.................................53
CHAPTER2 Random Variables...............................................................57
2.1 Introduction..........................................................................57
2.2 DefinitionofaRandomVariable..........................................57
2.3 EventsDefinedbyRandomVariables..................................58
2.4 DistributionFunctions..........................................................59
2.5 DiscreteRandomVariables.................................................61
2.5.1 ObtainingthePMFfromtheCDF............................65
2.6 ContinuousRandomVariables............................................67
2.7 ChapterSummary................................................................72
2.8 Problems..............................................................................73
Section2.4 DistributionFunctions...................................73
Section2.5 DiscreteRandomVariables...........................75
Section2.6 ContinuousRandomVariables......................77
CHAPTER3 Moments ofRandom Variables..........................................81
3.1 Introduction..........................................................................81
3.2 Expectation...........................................................................82
3.3 ExpectationofNonnegativeRandomVariables..................84
3.4 MomentsofRandomVariablesandtheVariance...............86
3.5 ConditionalExpectations......................................................95
3.6 TheMarkovInequality..........................................................96
3.7 TheChebyshevInequality....................................................97
Contents vii
3.8 ChapterSummary................................................................98
3.9 Problems..............................................................................98
Section3.2 ExpectedValues.............................................98
Section3.4 MomentsofRandomVariablesandthe
Variance........................................................100
Section3.5 ConditionalExpectations.............................101
Sections3.6and3.7 MarkovandChebyshev
Inequalities....................................102
CHAPTER4 Special Probability Distributions.......................................103
4.1 Introduction........................................................................103
4.2 TheBernoulliTrialandBernoulliDistribution.................103
4.3 BinomialDistribution.........................................................105
4.4 GeometricDistribution.......................................................108
4.4.1 CDFoftheGeometricDistribution........................111
4.4.2 ModifiedGeometricDistribution............................111
4.4.3 “Forgetfulness”PropertyoftheGeometric
Distribution.............................................................112
4.5 PascalDistribution.............................................................113
4.5.1 DistinctionBetweenBinomialandPascal
Distributions...........................................................117
4.6 HypergeometricDistribution.............................................118
4.7 PoissonDistribution...........................................................122
4.7.1 PoissonApproximationoftheBinomial
Distribution.............................................................123
4.8 ExponentialDistribution.....................................................124
4.8.1 “Forgetfulness”PropertyoftheExponential
Distribution.............................................................126
4.8.2 RelationshipbetweentheExponentialand
PoissonDistributions.............................................127
4.9 ErlangDistribution.............................................................128
4.10 UniformDistribution..........................................................133
4.10.1 TheDiscreteUniformDistribution......................134
4.11 NormalDistribution...........................................................135
4.11.1 NormalApproximationoftheBinomial
Distribution...........................................................138
4.11.2 TheErrorFunction...............................................139
4.11.3 TheQ-Function.....................................................140
4.12 TheHazardFunction..........................................................141
4.13 TruncatedProbabilityDistributions...................................143
4.13.1 TruncatedBinomialDistribution..........................145
4.13.2 TruncatedGeometricDistribution.......................145
viii Contents
4.13.3 TruncatedPoissonDistribution...........................145
4.13.4 TruncatedNormalDistribution............................146
4.14 ChapterSummary..............................................................146
4.15 Problems............................................................................147
Section4.3 BinomialDistribution...................................147
Section4.4 GeometricDistribution.................................151
Section4.5 PascalDistribution.......................................152
Section4.6 HypergeometricDistribution.......................153
Section4.7 PoissonDistribution.....................................154
Section4.8 ExponentialDistribution..............................154
Section4.9 ErlangDistribution.......................................156
Section4.10 UniformDistribution..................................157
Section4.11 NormalDistribution...................................158
CHAPTER5 MultipleRandom Variables...............................................159
5.1 Introduction........................................................................159
5.2 JointCDFsofBivariateRandomVariables.......................159
5.2.1 PropertiesoftheJointCDF...................................159
5.3 DiscreteBivariateRandomVariables................................160
5.4 ContinuousBivariateRandomVariables...........................163
5.5 DeterminingProbabilitiesfromaJointCDF.....................165
5.6 ConditionalDistributions...................................................168
5.6.1 ConditionalPMFforDiscreteBivariate
RandomVariables..................................................168
5.6.2 ConditionalPDFforContinuousBivariate
RandomVariables..................................................169
5.6.3 ConditionalMeansandVariances..........................170
5.6.4 SimpleRuleforIndependence..............................171
5.7 CovarianceandCorrelationCoefficient.............................172
5.8 MultivariateRandomVariables..........................................176
5.9 MultinomialDistributions..................................................177
5.10 ChapterSummary..............................................................179
5.11 Problems............................................................................179
Section5.3 DiscreteBivariateRandomVariables.........179
Section5.4 ContinuousBivariateRandomVariables.....180
Section5.6 ConditionalDistributions.............................182
Section5.7 CovarianceandCorrelationCoefficient......183
Section5.9 MultinomialDistributions............................183
CHAPTER6 Functions ofRandom Variables........................................185
6.1 Introduction........................................................................185
6.2 FunctionsofOneRandomVariable...................................185
6.2.1 LinearFunctions....................................................185
Contents ix
6.2.2 PowerFunctions....................................................187
6.3 ExpectationofaFunctionofOneRandomVariable.........188
6.3.1 MomentsofaLinearFunction...............................188
6.3.2 ExpectedValueofaConditionalExpectation........189
6.4 SumsofIndependentRandomVariables..........................189
6.4.1 MomentsoftheSumofRandomVariables..........196
6.4.2 SumofDiscreteRandomVariables.......................197
6.4.3 SumofIndependentBinomialRandom
Variables.................................................................200
6.4.4 SumofIndependentPoissonRandomVariables..201
6.4.5 TheSparePartsProblem......................................201
6.5 MinimumofTwoIndependentRandomVariables............204
6.6 MaximumofTwoIndependentRandomVariables...........205
6.7 ComparisonoftheInterconnectionModels......................207
6.8 TwoFunctionsofTwoRandomVariables.........................209
6.8.1 ApplicationoftheTransformationMethod...........210
6.9 LawsofLargeNumbers....................................................212
6.10 TheCentralLimitTheorem...............................................214
6.11 OrderStatistics..................................................................215
6.12 ChapterSummary..............................................................219
6.13 Problems............................................................................219
Section6.2 FunctionsofOneRandomVariable.............219
Section6.4 SumsofRandomVariables.........................220
Sections6.4and6.5 MaximumandMinimumof
IndependentRandomVariables....221
Section6.8 TwoFunctionsofTwoRandomVariables...222
Section6.10 TheCentralLimitTheorem.......................222
Section6.11 OrderStatistics..........................................223
CHAPTER7 Transform Methods...........................................................225
7.1 Introduction........................................................................225
7.2 TheCharacteristicFunction..............................................225
7.2.1 Moment-GeneratingPropertyofthe
CharacteristicFunction..........................................226
7.2.2 SumsofIndependentRandomVariables..............227
7.2.3 TheCharacteristicFunctionsofSome
Well-KnownDistributions......................................228
7.3 TheS-Transform.................................................................231
7.3.1 Moment-GeneratingPropertyofthes-Transform231
7.3.2 Thes-TransformofthePDFoftheSumof
IndependentRandomVariables.............................232
7.3.3 Thes-TransformsofSomeWell-KnownPDFs.....232