Table Of ContentTHIRD EDITION
FUNDAMENTALS
OF PROBABILITY
WITH STOCHASTIC PROCESSES
SAEED GHAHRAMANI
Western New England College
UpperSaddleRiver,NewJersey07458
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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
Description:Presenting probability in a natural way, this book uses interesting, carefully selected instructive examples that explain the theory, definitions, theorems, and methodology. Fundamentals of Probability has been adopted by the American Actuarial Society as one of its main references for the mathemati