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probability and StatisticsThe Science of UncertaintySecond EditionMichael J. Evans and Je rey S. RosenthalUniversity of TorontoContentsPrefaceix1 probability probability : A Measure of Uncertainty .. Why Do We Need probability Theory?.. probability VennDiagramsandSubsets .. Properties of probability Models .. Uniform probability on Finite Spaces .. Conditional probability and Independence .. Conditional probability .. IndependenceofEvents .. Continuity FurtherProofs(Advanced).

to probability and statistics with mathematical content. Where possible, we provide mathematical details, and it is expected that students are seeking to gain some mastery

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1 probability and StatisticsThe Science of UncertaintySecond EditionMichael J. Evans and Je rey S. RosenthalUniversity of TorontoContentsPrefaceix1 probability probability : A Measure of Uncertainty .. Why Do We Need probability Theory?.. probability VennDiagramsandSubsets .. Properties of probability Models .. Uniform probability on Finite Spaces .. Conditional probability and Independence .. Conditional probability .. IndependenceofEvents .. Continuity FurtherProofs(Advanced).

2 312 Random Variables and CdfsofDiscreteDistributions .. CdfsofAbsolutelyContinuousDistributions .. Distributions Neither Discrete Nor Continuous (Advanced) .. One-DimensionalChangeofVariable .. TheDiscreteCase .. Joint probability Functions .. JointDensityFunctions .. ConditioningandIndependence .. Conditioning on Discrete Random Variables .. Conditioning on Continuous Random Variables .. IndependenceofRandomVariables.

3 Multidimensional Change of TheDiscreteCase .. TheContinuousCase(Advanced) .. Simulating probability Distributions .. SimulatingDiscreteDistributions .. (Advanced) .. 1253 Variance,Covariance,andCorrelation .. CharacteristicFunctions(Advanced).. ConditionalExpectation .. Conditional Variance (Advanced) .. Inequalities .. Jensen sInequality(Advanced) .. GeneralExpectations(Advanced) .. FurtherProofs(Advanced) .. 1944 Sampling Distributions and SamplingDistributions.

4 Convergence in probability .. Convergence with probability 1 .. TheStrongLawofLargeNumbers .. ConvergenceinDistribution .. TheCentralLimitTheorem .. The Central Limit Theorem and Assessing Error .. NormalDistributionTheory .. TheChi-SquaredDistribution .. FurtherProofs(Advanced) .. 2465 Statistical WhyDoWeNeedStatistics?.. Inference Using a probability Model .. FinitePopulations .. SimpleRandomSampling .. SomeBasicInferences .. PlottingData.

5 TypesofInference .. 2896 Likelihood TheLikelihoodFunction .. Maximum Likelihood Estimation .. The Multidimensional Case (Advanced).. StandardErrors,Bias,andConsistency .. ConfidenceIntervals .. TestingHypothesesandP-Values .. Sample-Size Calculations: Sample-SizeCalculations:Power .. Distribution-FreeMethods .. MethodofMoments .. The Sign Statistic and Inferences about Quantiles .. LargeSampleBehavioroftheMLE(Advanced).. 3647 Bayesian Sampling from the Posterior Using Gibbs Sampling(Advanced).

6 ConjugatePriors .. EmpiricalBayes .. FurtherProofs(Advanced) .. Derivation of the Posterior Distribution for the Location-ScaleNormalModel .. Derivation ofJ( ( 0, ))for the Location-Scale Normal .. 4318 Optimal Optimal Unbiased Estimation .. The Rao Blackwell Theorem and Rao Blackwellization .. Completeness and the Lehmann Scheff Theorem .. The Cramer Rao Inequality (Advanced) .. Optimal Hypothesis Testing .. ThePowerFunctionofaTest .. TheNeyman PearsonTheorem.

7 LikelihoodRatioTests(Advanced) .. Optimal Bayesian Inferences .. DecisionTheory(Advanced).. FurtherProofs(Advanced) .. 4739 Model CheckingtheSamplingModel .. Residual and probability PredictionandCross-Validation .. WhatDoWeDoWhenaModelFails?.. Checking for Prior Data The Problem with Multiple Checks .. 50910 Relationships Among .. The Cause EffectRelationshipsandExperiments .. RandomPredictor .. BayesianFormulation .. Quantitative Response and Predictors.

8 TheMethodofLeastSquares .. TheSimpleLinearRegressionModel .. BayesianSimpleLinearModel(Advanced).. The Multiple Linear Regression Model (Advanced) .. Quantitative Response and Categorical Predictors .. OneCategoricalPredictor(One-WayANOVA) .. RepeatedMeasures(PairedComparisons).. TwoCategoricalPredictors(Two-WayANOVA) .. RandomizedBlocks .. One Categorical and One Quantitative Categorical Response and Quantitative Predictors .. (Advanced) .. 60711 Advanced Topic Stochastic TheDistributionoftheFortune.

9 TheGambler sRuinProblem .. StationaryDistributions .. Markov Chain Limit Theorem .. TheMetropolis .. Definition of a Martingale .. ExpectedValues .. Brownian Motion as a Limit .. DiffusionsandStockPrices .. 668 Appendices675A Mathematical Derivatives .. Matrix Multiplication .. MultivariableIntegrals .. 679B 699C Common 706D Chi-Squared Distribution Quantiles .. Quantiles .. Binomial Distribution Probabilities.

10 724E Answers to Odd-Numbered Exercises729 Index750 PrefaceThis book is an introductory text on probability and statistics, targeting students whohave studied one year of calculus at the University level and are seeking an introductionto probability and statistics with mathematical content. Where possible, we providemathematical details, and it is expected that students are seeking to gain some masteryover these, as well as to learn how to conduct data analyses. All the usual method-ologies covered in a typical introductory course are introduced, as well as some of thetheory that serves as their text can be used with or without a statistical computer package.

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