Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

796 Pages·2012·34.07 MB·English

by  John J. Shynk

PROBABILITY, RANDOM
VARIABLES, AND RANDOM
PROCESSES

PROBABILITY, RANDOM
VARIABLES, AND RANDOM
PROCESSES
Theory and Signal Processing Applications
JOHN J. SHYNK
DepartmentofElectricalandComputerEngineering
UniversityofCalifornia,SantaBarbara,CA,USA
AJOHNWILEY&SONS,INC.,PUBLICATION
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Shynk,JohnJoseph.
Probability,randomvariables,andrandomprocesses:theoryandsignalprocessingapplications/JohnJ.Shynk.
p.cm.
Includesindex.
ISBN978-0-470-24209-4(hardback)
1.Probabilities–Textbooks. 2.Stochasticprocesses–Textbooks. 3.Engineering–Statisticalmethods–Textbooks.
I.Title.
QA274.S532012
519.2–dc23
2012008186
PrintedintheUnitedStatesofAmerica
ISBN:9780470242094
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CONTENTS
PREFACE xxi
NOTATION xxv
1 OverviewandBackground 1
1.1 Introduction 1
1.1.1 Signals,SignalProcessing,andCommunications 3
1.1.2 Probability,RandomVariables,andRandomVectors 9
1.1.3 RandomSequencesandRandomProcesses 11
1.1.4 DeltaFunctions 16
1.2 DeterministicSignalsandSystems 19
1.2.1 ContinuousTime 20
1.2.2 DiscreteTime 25
1.2.3 Discrete-TimeFilters 29
1.2.4 State-SpaceRealizations 32
1.3 StatisticalSignalProcessingwithMATLAB® 35
1.3.1 RandomNumberGeneration 35
1.3.2 Filtering 38
Problems 39
FurtherReading 45
PARTI Probability,RandomVariables,andExpectation
2 ProbabilityTheory 49
2.1 Introduction 49
2.2 SetsandSampleSpaces 50
2.3 SetOperations 54
2.4 EventsandFields 58
2.5 SummaryofaRandomExperiment 64
2.6 MeasureTheory 64
2.7 AxiomsofProbability 68
2.8 BasicProbabilityResults 69
2.9 ConditionalProbability 71
2.10 Independence 73
vii
viii CONTENTS
2.11 Bayes’Formula 74
2.12 TotalProbability 76
2.13 DiscreteSampleSpaces 79
2.14 ContinuousSampleSpaces 83
2.15 NonmeasurableSubsetsofR 84
Problems 87
FurtherReading 90
3 RandomVariables 91
3.1 Introduction 91
3.2 FunctionsandMappings 91
3.3 DistributionFunction 96
3.4 ProbabilityMassFunction 101
3.5 ProbabilityDensityFunction 103
3.6 MixedDistributions 104
3.7 ParametricModelsforRandomVariables 107
3.8 ContinuousRandomVariables 109
3.8.1 GaussianRandomVariable(Normal) 110
3.8.2 Log-NormalRandomVariable 113
3.8.3 InverseGaussianRandomVariable(Wald) 114
3.8.4 ExponentialRandomVariable(One-Sided) 116
3.8.5 LaplaceRandomVariable(Double-SidedExponential) 119
3.8.6 CauchyRandomVariable 122
3.8.7 ContinuousUniformRandomVariable 124
3.8.8 TriangularRandomVariable 125
3.8.9 RayleighRandomVariable 127
3.8.10 RiceRandomVariable 129
3.8.11 GammaRandomVariable(Erlangforr ∈N) 131
3.8.12 BetaRandomVariable(Arcsineforα =β =1/2,PowerFunctionfor
β =1) 133
3.8.13 ParetoRandomVariable 136
3.8.14 WeibullRandomVariable 137
3.8.15 LogisticRandomVariable(Sigmoidfor{μ=0,α =1}) 139
3.8.16 ChiRandomVariable(Maxwell–Boltzmann,Half-Normal) 141
3.8.17 Chi-SquareRandomVariable 144
3.8.18 F-Distribution 147
3.8.19 Student’stDistribution 149
3.8.20 ExtremeValueDistribution(TypeI:Gumbel) 150
3.9 DiscreteRandomVariables 151
3.9.1 BernoulliRandomVariable 152
3.9.2 BinomialRandomVariable 154
3.9.3 GeometricRandomVariable(withSupportZ+orN) 157