Fundamentals of Probability, with Stochastic Processes

672 Pages·2004·4.347 MB·English

by  Saeed Ghahramani

THIRD EDITION
FUNDAMENTALS
OF PROBABILITY
WITH STOCHASTIC PROCESSES
SAEED GHAHRAMANI
Western New England College
UpperSaddleRiver,NewJersey07458
LibraryofCongressCataloging-in-PublicationData
Ghahramani,Saeed.
Fundamentalsofprobabilitywithstochasticprocesses/SaeedGhahramani.—3rdedition.
p. cm.
IncludesIndex.
ISBN: 0-13-145340-8
1. Probabilities. I.Title.
QA273.G4642005
519.2—dc22 2004048541
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C
ontents
! Preface xi
! 1 Axioms of Probability 1
1.1 Introduction 1
1.2 SampleSpaceandEvents 3
1.3 AxiomsofProbability 11
1.4 BasicTheorems 18
1.5 ContinuityofProbabilityFunction 27
1.6 Probabilities0and1 29
1.7 RandomSelectionofPointsfromIntervals 30
ReviewProblems 35
! 2 Combinatorial Methods 38
2.1 Introduction 38
2.2 CountingPrinciple 38
NumberofSubsetsofaSet 42
TreeDiagrams 42
2.3 Permutations 47
2.4 Combinations 53
2.5 Stirling’sFormula 70
ReviewProblems 71
! 3 Conditional Probability and Independence 75
3.1 ConditionalProbability 75
ReductionofSampleSpace 79
3.2 LawofMultiplication 85
3.3 LawofTotalProbability 88
3.4 Bayes’Formula 100
3.5 Independence 107
v
vi Contents
3.6 ApplicationsofProbabilitytoGenetics 126
Hardy-WeinbergLaw 130
Sex-LinkedGenes 132
ReviewProblems 136
Distribution Functions and
! 4 139
Discrete RandomVariables
4.1 RandomVariables 139
4.2 DistributionFunctions 143
4.3 DiscreteRandomVariables 153
4.4 ExpectationsofDiscreteRandomVariables 159
4.5 VariancesandMomentsofDiscreteRandomVariables 175
Moments 181
4.6 StandardizedRandomVariables 184
ReviewProblems 185
! 5 Special Discrete Distributions 188
5.1 BernoulliandBinomialRandomVariables 188
ExpectationsandVariancesofBinomialRandomVariables 194
5.2 PoissonRandomVariable 201
PoissonasanApproximationtoBinomial 201
PoissonProcess 206
5.3 OtherDiscreteRandomVariables 215
GeometricRandomVariable 215
NegativeBinomialRandomVariable 218
HypergeometricRandomVariable 220
ReviewProblems 228
! 6 Continuous RandomVariables 231
6.1 ProbabilityDensityFunctions 231
6.2 DensityFunctionofaFunctionofaRandomVariable 240
6.3 ExpectationsandVariances 246
ExpectationsofContinuousRandomVariables 246
VariancesofContinuousRandomVariables 252
ReviewProblems 258
Contents vii
! 7 Special Continuous Distributions 261
7.1 UniformRandomVariable 261
7.2 NormalRandomVariable 267
CorrectionforContinuity 270
7.3 ExponentialRandomVariables 284
7.4 GammaDistribution 292
7.5 BetaDistribution 297
7.6 SurvivalAnalysisandHazardFunction 303
ReviewProblems 308
! 8 Bivariate Distributions 311
8.1 JointDistributionofTwoRandomVariables 311
JointProbabilityMassFunctions 311
JointProbabilityDensityFunctions 315
8.2 IndependentRandomVariables 330
IndependenceofDiscreteRandomVariables 331
IndependenceofContinuousRandomVariables 334
8.3 ConditionalDistributions 343
ConditionalDistributions: DiscreteCase 343
ConditionalDistributions: ContinuousCase 349
8.4 TransformationsofTwoRandomVariables 356
ReviewProblems 365
! 9 Multivariate Distributions 369
9.1 JointDistributionofn>2RandomVariables 369
JointProbabilityMassFunctions 369
JointProbabilityDensityFunctions 378
RandomSample 382
9.2 OrderStatistics 387
9.3 MultinomialDistributions 394
ReviewProblems 398
! 10 More Expectations andVariances 400
10.1 ExpectedValuesofSumsofRandomVariables 400
PatternAppearance 407
10.2 Covariance 415
viii Contents
10.3 Correlation 429
10.4 ConditioningonRandomVariables 434
10.5 BivariateNormalDistribution 449
ReviewProblems 454
Sums of Independent Random
! 11 457
Variables and LimitTheorems
11.1 Moment-GeneratingFunctions 457
11.2 SumsofIndependentRandomVariables 468
11.3 MarkovandChebyshevInequalities 476
Chebyshev’sInequalityandSampleMean 480
11.4 LawsofLargeNumbers 486
ProportionversusDifferenceinCoinTossing 495
11.5 CentralLimitTheorem 498
ReviewProblems 507
! 12 Stochastic Processes 511
12.1 Introduction 511
12.2 MoreonPoissonProcesses 512
WhatIsaQueuingSystem? 523
PASTA:PoissonArrivalsSeeTimeAverage 525
12.3 MarkovChains 528
ClassificationsofStatesofMarkovChains 538
AbsorptionProbability 549
Period 552
Steady-StateProbabilities 554
12.4 Continuous-TimeMarkovChains 566
Steady-StateProbabilities 572
BirthandDeathProcesses 576
12.5 BrownianMotion 586
FirstPassageTimeDistribution 593
TheMaximumofaBrownianMotion 594
TheZerosofBrownianMotion 594
BrownianMotionwithDrift 597
GeometricBrownianMotion 598
ReviewProblems 602
Contents ix
! 13 Simulation 606
13.1 Introduction 606
13.2 SimulationofCombinatorialProblems 610
13.3 SimulationofConditionalProbabilities 614
13.4 SimulationofRandomVariables 617
13.5 MonteCarloMethod 626
! AppendixTables 630
! Answers to Odd-Numbered Exercises 634
! Index 645