Fundamentals of Probability: A First Course. Anirban DasGupta

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1 Fundamentals of Probability: A First Course Anirban DasGupta

2 Contents 1 Introducing Probability ExperimentsandSampleSpaces Set Theory Notation and Axioms of Probability How to Interpret a Probability Calculating Probabilities ManualCounting GeneralCountingMethods InclusionExclusionFormula Bounds on the Probability of a Union Synopsis Exercises References The Birthday and the Matching Problem TheBirthdayProblem *Stirling sapproximation TheMatchingProblem Synopsis Exercises References Conditional Probability and Independence BasicFormulasandFirstExamples MoreAdvancedExamples IndependentEvents BayesTheorem Synopsis Exercises Integer Valued and Discrete Random Variables MassFunction CDFandMedianofaRandomVariable FunctionsofaRandomVariable Independence of Random Variables ExpectedValueofaDiscreteRandomVariable I

3 4.4 BasicPropertiesofExpectations Illustrative Examples Using Indicator Variables to Calculate Expectations The Tail Sum Method for Calculating Expectations Variance, Moments, and Basic Inequalities Illustrative Examples Variance of a Sum of Independent Random Variables Utility of µ, σ assummaries Chebyshev s Inequality and Weak Law of Large Numbers *BetterInequalities Other Fundamental Moment Inequalities *ApplyingMomentInequalities TruncatedDistributions Synopsis Exercises References Generating Functions GeneratingFunctions Moment Generating Functions and Cumulants Cumulants Synopsis Exercises References Standard Discrete Distributions IntroductiontoSpecialDistributions DiscreteUniformDistribution BinomialDistribution Geometric and Negative Binomial Distribution HypergeometricDistribution PoissonDistribution Mean Absolute Deviation and the Mode PoissonApproximationtoBinomial MiscellaneousPoissonApproximations Benford slaw II

4 6.10 DistributionofSumsandDifferences *DistributionofDifferences DiscreteDoesNotMeanIntegerValued Synopsis Exercises References Continuous Random Variables TheDensityFunctionandtheCDF Quantiles Generating New Distributions from Old Normal and Other Symmetric Unimodal Densities Functions of a Continuous Random Variable QuantileTransformation Cauchydensity ExpectationofFunctionsandMoments The Tail Probability Method for Calculating Expectations SurvivalandHazardRate *MomentsandtheTail Moment Generating Function and Fundamental Tail Inequalities *Chernoff-BernsteinInequality *Lugosi simprovedinequality Jensen and Other Moment Inequalities and a Paradox Synopsis Exercises References Some Special Continuous Distributions UniformDistribution ExponentialandWeibullDistributions GammaandInverseGammaDistributions BetaDistribution ExtremeValueDistributions * Exponential Density and the Poisson Process Synopsis Exercises III

5 8.9 References Normal Distribution DefinitionandBasicProperties WorkingwithaNormalTable Additional Examples and the Lognormal Density SumsofIndependentNormalVariables Mills Ratio and Approximations for the Standard Normal CDF Synopsis Exercises References Normal Approximations and Central Limit Theorem SomeMotivatingExamples CentralLimitTheorem NormalApproximationtoBinomial ContinuityCorrection ANewRuleofThumb ExamplesoftheGeneralCLT NormalApproximationtoPoissonandGamma Convergence of Densities and Higher Order Approximations *RefinedApproximations Practical Recommendations for Normal Approximations Synopsis Exercises References Multivariate Discrete Distributions Bivariate Joint Distributions and Expectations of Functions Conditional Distributions and Conditional Expectations Examples on Conditional Distributions and Expectations Using Conditioning to Evaluate Mean and Variance CovarianceandCorrelation MultivariateCase JointMGF MultinomialDistribution IV

6 11.6 Synopsis Exercises Multidimensional Densities JointDensityFunctionandItsRole ExpectationofFunctions BivariateNormal Conditional Densities and Expectations Examples on Conditional Densities and Expectations Bivariate Normal Conditional Distributions OrderStatistics BasicDistributionTheory * More Advanced Distribution Theory Synopsis Exercises References Convolutions and Transformations ConvolutionsandExamples Products and Quotients and the t and F Distribution Transformations ApplicationsofJacobianFormula PolarCoordinatesinTwoDimensions Synopsis Exercises References Markov Chains and Applications NotationandBasicDefinitions Chapman-Kolmogorov Equation CommunicatingClasses Gambler sruin FirstPassage,RecurrenceandTransience Long Run Evolution and Stationary Distributions Synopsis Exercises V

7 14.9 References Urn Models in Physics and Genetics Stirling Numbers and Their Basic Properties UrnModelsinQuantumMechanics *PoissonApproximations Pólya surn Pólya-EggenbergerDistribution * de Finetti s Theorem and PólyaUrns UrnModelsinGenetics Wright-FisherModel TimeuntilAlleleUniformity MutationandHoppe surn *TheEwensSamplingFormula Synopsis Exercises References Appendix I: Supplementary Homework and Practice Problems WordProblems True-FalseProblems Appendix II GlossaryofSymbols FormulaSummaries Moments and MGFs of Common Distributions Useful Mathematical Formulas UsefulCalculusFacts Tables NormalTable PoissonTable VI