概率论基础教程

1 CombinatorialAnalysis

1.1 Introduction

1.2 TheBasicPrincipleofCounting

1.3 Permutations

1.4 Combinations

1.5 MultinomialCoefficients

1.6 TheNumberofIntegerSolutlonsofEquations

Summary

Problems

TheoreticalExercises

Self-TestProblemsandExercises

2 AxiomsofProbability

2.1 Introduction

2.2 SampleSpaceandEvents

2.3 AxiomsofProbability

2.4 SomeSimplePropositions

2.5 SampleSpaceHavingEquallyLikelyOutcomes 33

2.6 ProbabilityasaContinuousSetFunction

2.7 ProbabilityasaMeasureofBelief

Summary

Problems

TheoreticalExercises

Self-TestProblemsandExercises

3 ConditionalProbabilityandIndependence

3.1 Introduction

3.2 ConditionalProbabilities

3.3 BayessFormula

3.4 IndependentEvents

3.5 P(·|F)IsaProbability

Summary

Problems

TheoreticalExercises

Self-TestProblemsandExercises

4 RandomVariables

4.1 RandomVariables

4.2 DiscreteRandomVariables

4.3 ExpectedValue

4.4 ExpectationofaFunctionofaRandomVariable

4.5 Variance

4.6 TheBernoulhandBinomialRandomVariables

4.6.1 PropertiesofBinomialRandomVariables

4.6.2 ComputingtheBinomialDistributionFunction

4.7 ThePoissonRandomVariable

4.7.1 ComputingthePoissonDistributionFunction

4.8 OtherDiscreteProbabilityDistributions

4.8.1 TheGeometricRandomVariable

4.8.2 TheNegativeBinomialRandomVariable

4.8.3 TheHypergeometricRandomVariable

4.8.4 TheZeta(orZipf)Distribution

4.9 ExpectedValueofSumsofRandomVariables

4.10 PropertiesoftheCumulativeDistributionFunction

Summary

Problems

TheoreticalExercises

Self-TestProblemsandExercises

5 ContinuousRandomVariables

51 Introduction

5.2 ExpectationandVarianceofContinuousRandomVariables

5.3 TheUniformRandomVariable

5.4 NormalRandomVariables

5.4.1 TheNormalApproximationtotheBinomialDistribution

5.5 ExponentialRandomVariables

5.5.1 HazardRateFunctions

5.6 OtherContinuousDistributions

5.6.1 TheGammaDlstrlbutlon

5.6.2 TheWeibullDlStrlbutlon

5.6.3 TheCauchyDistribution

5.6.4 TheBetaDlStrlbutlon

5.7 TheDistributionofaFunctionofaRandomVariable

Summary

Problems

TheoreticalExercises

Self-TestProblemsandExercises

6 JointlyDistributedRandomVariables

6.1 JointDistributionFunctions

6.2 IndependentRandomVariables

6.3 SumsofIndependentRandomVariables

6.3.1 IdenticallyDistributedUniformRandomVariables

6.3.2 GammaRandomVariables

6.3.3 NormalRandomVariables

6.3.4 PolssonandBinomialRandomVariables

635 GeometricRandomVariables

6.4 ConditionalDistribution：DiscreteCase

6.5 ConditionalDistribution：ContinuousCase

66 OrderStatistics

6.7 JointProbabilityDistributionofFunctionsofRandomVariables

6.8 ExciaanzeaoleRandomVariables

Summary

Problems

TheoreticalExercises

SelfTestProblemsandExercises

7 PropertiesofExpectation

7.1 Introduction

7.2 ExpectationofSumsofRandomVariablviatheProbabilisticMethod

7.2.2 TheMaximum-MinimumsIdentity

7.3 MomentsoftheNumberofEventsthatOccur

7.4 Covariance，VarianceofSums，andCorrelations

7.5 ConditionalExpectation

7.5.1 Definitions

7.5.2 ComputingExpectationsbyConditioning

7.5.3 ComputingProbabilitiesbyConditioning

7.5.4 ConditionalVariance

7.6 ConditionalExpectationandPrediction

7.7 MomentGeneratingFunctions

7.7.1 JointMomentGeneratingFunctions

7.8 AddltlonaproprietariesofNormalRandomVariables

7.8.1 TheMultivariateNormalDlstrlbution

7.8.2 TheJointDistributionoftheSampleMeanandSampleVariance

7.9 GeneralDefinitionofExpectation

Summary

Problems

TheoreticalExercises

Self-TestProblemsandExercises

8 LimitTheorems

8.1 Introduction

8.2 ChebyshevsInequalityandtheWeakLawofLargeNumbers

8.3 TheCentralLimitTheorem

8.4 TheStrongLawofLargeNumbers

8.5 OtherInequamles

8.6 BoundingtheErrorProbabilityWhenApproximatingaSumofIndependentBernoulliRandomVariablesbyaPoissonRandomVariable

Summary

Problems

TheoreticalExercises

Self-TestProblemsandExercises

9 AdditionalTopicsinProbability

9.1 ThePoissonProcess

9.2 MarkovChains

9.3 Surprise，Uncertainty，andEntropy

9.4 CodingTheoryandEntropy

Summary

ProblemsandTheoreticalExercises

Self-TestProblemsandExercises

References

10 Simulation

10.1 Introduction

10.2 GeneralTechniquesforSimulatingContinuousRandomVariables

10.2.1 TheInverseTransformationMethod

10.2.2 TheRejectionMethod

10.3 SimulatingfromDiscreteDistributions

10.4 VarianceReductionTechniques

10.4.1 UseofAntitheticVariables

10.4.2 VarianceReductionbyConditioning

10.4.3 ControlVariates

Summary

Problems

Self-TestProblemsandExercises

Reference

AnswerstoSelectedProblems

SolutionstoSelf-TestProblemsandExercises

Index

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