S.M.Ross，第九版。目录见下。PDF格式。

Content

1. Introduction to Probability Theory 1
1.1. Introduction 1
1.2. SampleSpaceandEvents 1
1.3. ProbabilitiesDefinedonEvents 4
1.4. ConditionalProbabilities 7
1.5. IndependentEvents 10
1.6. Bayes’Formula 12
Exercises 15
References 21
2. Random Variables 23
2.1. RandomVariables 23
2.2. DiscreteRandom Variables 27
2.2.1. TheBernoulliRandom Variable 28
2.2.2. TheBinomialRandomVariable 29
2.2.3. TheGeometricRandomVariable 31
2.2.4. ThePoissonRandomVariable 32
2.3. ContinuousRandomVariables 34
2.3.1. TheUniform RandomVariable 35
2.3.2. ExponentialRandomVariables 36
2.3.3. GammaRandomVariables 37
2.3.4. NormalRandomVariables 37
v

vi Contents
2.4. ExpectationofaRandomVariable 38
2.4.1. TheDiscreteCase 38
2.4.2. TheContinuousCase 41
2.4.3. ExpectationofaFunctionofaRandomVariable 43
2.5. JointlyDistributedRandomVariables 47
2.5.1. JointDistributionFunctions 47
2.5.2. IndependentRandomVariables 51
2.5.3. CovarianceandVarianceofSumsof RandomVariables 53
2.5.4. JointProbabilityDistributionof FunctionsofRandom
Variables 61
2.6. MomentGeneratingFunctions 64
2.6.1. TheJointDistributionoftheSampleMeanandSample
Variancefrom aNormalPopulation 74
2.7. LimitTheorems 77
2.8. StochasticProcesses 83
Exercises 85
References 96
3. Conditional Probability and Conditional
Expectation 97
3.1. Introduction 97
3.2. TheDiscreteCase 97
3.3. TheContinuousCase 102
3.4. ComputingExpectationsbyConditioning 105
3.4.1. ComputingVariancesbyConditioning 117
3.5. ComputingProbabilitiesbyConditioning 120
3.6. SomeApplications 137
3.6.1. AList Model 137
3.6.2. ARandomGraph 139
3.6.3. Uniform Priors, Polya’sUrnModel,and
Bose–EinsteinStatistics 147
3.6.4. MeanTimefor Patterns 151
3.6.5. Thek-RecordValuesof DiscreteRandomVariables 155
3.7. AnIdentityfor CompoundRandom Variables 158
3.7.1. PoissonCompoundingDistribution 161
3.7.2. BinomialCompoundingDistribution 163
3.7.3. ACompoundingDistributionRelatedtotheNegative
Binomial 164
Exercises 165

Contents vii
4. Markov Chains 185
4.1. Introduction 185
4.2. Chapman–KolmogorovEquations 189
4.3. ClassificationofStates 193
4.4. LimitingProbabilities 204
4.5. SomeApplications 217
4.5.1. TheGambler’sRuinProblem 217
4.5.2. AModelfor AlgorithmicEfficiency 221
4.5.3. Usinga RandomWalktoAnalyzeaProbabilisticAlgorithm
for theSatisfiabilityProblem 224
4.6. MeanTimeSpentinTransientStates 230
4.7. BranchingProcesses 233
4.8. TimeReversibleMarkovChains 236
4.9. MarkovChainMonteCarloMethods 247
4.10. MarkovDecisionProcesses 252
4.11. HiddenMarkovChains 256
4.11.1. PredictingtheStates 261
Exercises 263
References 280
5. The Exponential Distribution and the Poisson
Process 281
5.1. Introduction 281
5.2. TheExponentialDistribution 282
5.2.1. Definition 282
5.2.2. PropertiesoftheExponentialDistribution 284
5.2.3. FurtherPropertiesoftheExponentialDistribution 291
5.2.4. Convolutionsof ExponentialRandomVariables 298
5.3. ThePoissonProcess 302
5.3.1. CountingProcesses 302
5.3.2. Definitionof thePoissonProcess 304
5.3.3. InterarrivalandWaitingTimeDistributions 307
5.3.4. FurtherPropertiesofPoissonProcesses 310
5.3.5. ConditionalDistributionof theArrivalTimes 316
5.3.6. EstimatingSoftwareReliability 328
5.4. GeneralizationsofthePoissonProcess 330
5.4.1. NonhomogeneousPoissonProcess 330
5.4.2. CompoundPoissonProcess 337
5.4.3. Conditionalor MixedPoissonProcesses 343

viii Contents
Exercises 346
References 364
6. Continuous-Time Markov Chains 365
6.1. Introduction 365
6.2. Continuous-TimeMarkovChains 366
6.3. Birth andDeathProcesses 368
6.4. TheTransitionProbabilityFunction P
(t) 375
ij
6.5. LimitingProbabilities 384
6.6. TimeReversibility 392
6.7. Uniformization 401
6.8. ComputingtheTransitionProbabilities 404
Exercises 407
References 415
7. Renewal Theory and Its Applications 417
7.1. Introduction 417
7.2. Distributionof N(t) 419
7.3. LimitTheoremsandTheirApplications 423
7.4. RenewalReward Processes 433
7.5. RegenerativeProcesses 442
7.5.1. AlternatingRenewalProcesses 445
7.6. Semi-MarkovProcesses 452
7.7. TheInspectionParadox 455
7.8. ComputingtheRenewalFunction 458
7.9. ApplicationstoPatterns 461
7.9.1. Patternsof DiscreteRandomVariables 462
7.9.2. TheExpectedTimetoaMaximalRunof DistinctValues 469
7.9.3. IncreasingRuns ofContinuousRandomVariables 471
7.10. TheInsuranceRuinProblem 473
Exercises 479
References 492
8. Queueing Theory 493
8.1. Introduction 493
8.2. Preliminaries 494
8.2.1. Cost Equations 495
8.2.2. Steady-StateProbabilities 496

Contents ix
8.3. ExponentialModels 499
8.3.1. A Single-ServerExponentialQueueingSystem 499
8.3.2. A Single-ServerExponentialQueueingSystem
HavingFiniteCapacity 508
8.3.3. A ShoeshineShop 511
8.3.4. A QueueingSystem withBulkService 514
8.4. Networkof Queues 517
8.4.1. OpenSystems 517
8.4.2. ClosedSystems 522
8.5. TheSystem M/G/1 528
8.5.1. Preliminaries:WorkandAnotherCostIdentity 528
8.5.2. ApplicationofWork to M/G/1 529
8.5.3. BusyPeriods 530
8.6. Variationsonthe M/G/1 531
8.6.1. The M/G/1 withRandom-SizedBatchArrivals 531
8.6.2. PriorityQueues 533
8.6.3. An M/G/1 OptimizationExample 536
8.6.4. The M/G/1 QueuewithServerBreakdown 540
8.7. TheModel G/M/1 543
8.7.1. The G/M/1 BusyandIdlePeriods 548
8.8. A FiniteSourceModel 549
8.9. MultiserverQueues 552
8.9.1. Erlang’sLoss System 553
8.9.2. The M/M/k Queue 554
8.9.3. The G/M/k Queue 554
8.9.4. The M/G/k Queue 556
Exercises 558
References 570
9. Reliability Theory 571
9.1. Introduction 571
9.2. StructureFunctions 571
9.2.1. MinimalPathandMinimalCutSets 574
9.3. ReliabilityofSystemsofIndependentComponents 578
9.4. BoundsontheReliabilityFunction 583
9.4.1. Methodof InclusionandExclusion 584
9.4.2. SecondMethodfor ObtainingBoundson r(p) 593
9.5. SystemLife asaFunctionofComponentLives 595
9.6. ExpectedSystemLifetime 604
9.6.1. AnUpperBoundontheExpectedLife ofa
ParallelSystem 608

x Contents
9.7. SystemswithRepair 610
9.7.1. ASeriesModelwithSuspendedAnimation 615
Exercises 617
References 624
10. Brownian Motion and Stationary Processes 625
10.1. BrownianMotion 625
10.2. HittingTimes,MaximumVariable,andtheGambler’sRuin
Problem 629
10.3. VariationsonBrownianMotion 631
10.3.1. BrownianMotionwithDrift 631
10.3.2. GeometricBrownianMotion 631
10.4. PricingStockOptions 632
10.4.1. AnExampleinOptionsPricing 632
10.4.2. TheArbitrageTheorem 635
10.4.3. TheBlack-ScholesOptionPricingFormula 638
10.5. WhiteNoise 644
10.6. GaussianProcesses 646
10.7. StationaryandWeaklyStationaryProcesses 649
10.8. HarmonicAnalysisof WeaklyStationaryProcesses 654
Exercises 657
References 662
11. Simulation 663
11.1. Introduction 663
11.2. GeneralTechniquesfor SimulatingContinuousRandom
Variables 668
11.2.1. TheInverseTransformationMethod 668
11.2.2. TheRejectionMethod 669
11.2.3. TheHazardRateMethod 673
11.3. SpecialTechniquesfor SimulatingContinuousRandom
Variables 677
11.3.1. TheNormalDistribution 677
11.3.2. TheGammaDistribution 680
11.3.3. TheChi-SquaredDistribution 681
11.3.4. TheBeta (n,m) Distribution 681
11.3.5. TheExponentialDistribution—TheVonNeumann
Algorithm 682
11.4. Simulatingfrom DiscreteDistributions 685
11.4.1. TheAliasMethod 688

Contents xi
11.5. StochasticProcesses 692
11.5.1. SimulatingaNonhomogeneousPoissonProcess 693
11.5.2. SimulatingaTwo-DimensionalPoissonProcess 700
11.6. VarianceReductionTechniques 703
11.6.1. UseofAntitheticVariables 704
11.6.2. VarianceReductionbyConditioning 708
11.6.3. ControlVariates 712
11.6.4. ImportanceSampling 714
11.7. DeterminingtheNumberofRuns 720
11.8. CouplingfromthePast 720
Exercises 723
References 731

论坛新手，望指点。