概率论基础教程

1CombinatorialAnalysis

1.1Introduction

1.2TheBasicPrincipleofCounting

1.3Permutations

1.4Combinations

1.5MultinomialCoefficients

1.6TheNumberofIntegerSolutionsofEquations*

Summary

Problems

TheoreticalExercises

Self-TestProblemsandExercises

2AxiomsofProbability

2.1Introduction

2.2SampleSpaceandEvents

2.3AxiomsofProbability

2.4SomeSimplePropositions

2.5SampleSpacesHavingEquallyLikelyOutcomes

2.6ProbabilityasaContinuousSetFunction*

2.7ProbabilityasaMeasureofBelief

Summary

Problems

TheoreticalExercises

Self-TestProblemsandExercises

3ConditionalProbabilityandIndependence

3.1Introduction

3.2ConditionalProbabilities

3.3BayesFormula

3.4IndependentEvents

3.5P(.|F)IsaProbability

Summary

Problems

TheoreticalExercises

Self-TestProblemsandExercises

4RandomVariables

4.1RandomVariables

4.2DiscreteRandomVariables

4.3ExpectedValue

4.4ExpectationofaFunctionofaRandomVariable

4.5Variance

4.6TheBernoulliandBinomialRandomVariables

4.6.1PropertiesofBinomialRandomVariables

4.6.2ComputingtheBinomialDistributionFunction

4.7ThePoissonRandomVariable

4.7.1ComputingthePoissonDistributionFunction

4.8OtherDiscreteProbabilityDistributions

4.8.1TheGeometricRandomVariable

4.8.2TheNegativeBinomialRandomVariable

4.8.3TheHypergeometricRandomVariable

4.8.4TheZeta(orZipf)Distribution

4.9PropertiesoftheCumulativeDistributionFunction

Summary

Problems

TheoreticalExercises

Self-TestProblemsandExercises

5ContinuousRandomVariables

5.1Introduction

5.2ExpectationandVarianceofContinuousRandomVariables

5.3TheUniformRandomVariable

5.4NormalRandomVariables

5.4.1TheNormalApproximationtotheBinomialDistribution

5.5ExponentialRandomVariables

5.5.1HazardRateFunctions

5.6OtherContinuousDistributions

5.6.1TheGammaDistribution

5.6.2TheWeibullDistribution

5.6.3TheCauchyDistribution

5.6.4TheBetaDistribution

5.7TheDistributionofaFunctionofaRandomVariable

Summary

Problems

TheoreticalExercises

Self-TestProblemsandExercises

6JointlyDistributedRandomVariables

6.1JointDistributionFunctions

6.2IndependentRandomVariables

6.3SumsofIndependentRandomVariables

6.4ConditionalDistributions:DiscreteCase

6.5ConditionalDistributions:ContinuousCase

6.6OrderStatistics*

6.7JointProbabilityDistributionofFunctionsofRandomVariables

6.8ExchangeableRandomVariables*

Summary

Problems

TheoreticalExercises

Self-TestProblemsandExercises

7PropertiesofExpectation

7.1Introduction

7.2ExpectationofSumsofRandomVariables

7.2.1ObtainingBoundsfromExpectationsviatheProbabilisticMethod*

7.2.2TheMaximum-MinimumsIdentity*

7.3MomentsoftheNumberofEventsthatOccur

7.4Covariance,VarianceofSums,andCorrelations

7.5ConditionalExpectation

7.5.1Definitions

7.5.2ComputingExpectationsbyConditioning

7.5.3ComputingProbabilitiesbyConditioning

7.5.4ConditionalVariance

7.6ConditionalExpectationandPrediction

7.7MomentGeneratingFunctions

7.7.1JointMomentGeneratingFunctions

7.8AdditionalPropertiesofNormalRandomVariables

7.8.1TheMultivariateNormalDistribution

7.8.2TheJointDistributionoftheSampleMeanandSampleVariance

7.9GeneralDefinitionofExpectation

Summary

Problems

TheoreticalExercises

Self-TestProblemsandExercises

8LimitTheorems

8.1Introduction

8.2ChebyshevsInequalityandtheWeakLawofLargeNumbers

8.3TheCentralLimitTheorem

8.4TheStrongLawofLargeNumbers

8.5OtherInequalities

8.6BoundingTheErrorProbability

Summary

Problems

TheoreticalExercises

Self-TestProblemsandExercises

9AdditionalTopicsinProbability

9.1ThePoissonProcess

9.2MarkovChains

9.3Surprise,Uncertainty,andEntropy

9.4CodingTheoryandEntropy

Summary

TheoreticalExercises

Self-TestProblemsandExercises

10Simulation

10.1Introduction

10.2GeneralTechniquesforSimulatingContinuousRandomVariables

10.2.1TheInverseTransformationMethod

10.2.2TheRejectionMethod

10.3SimulatingfromDiscreteDistributions

10.4VarianceReductionTechniques

10.4.1UseofAntitheticVariables

10.4.2VarianceReductionbyConditioning

10.4.3ControlVariates

Summary

Problems

Self-TestProblemsandExercises

APPENDICES

AAnswerstoSelectedProblems

BSolutionstoSelf-TestProblemsandExercises

Index