SOLUTIONS MANUAL - FOURTH EDITION (Classic Version) Morris DeGroot...

SOLUTIONS MANUAL

MARK SCHERVISH

PROBABILITY AND STATISTICS

FOURTH EDITION

(Classic Version) Morris DeGroot Mark Schervish 1 / 4

Contents Preface ................................................... vi

• Introduction to Probability1

1.2 Interpretations of Probability . . .................................. 1 1.4 SetTheory .............................................. 1 1.5 The Definition of Probability . . .................................. 3 1.6 FiniteSampleSpaces......................................... 6 1.7 CountingMethods .......................................... 7 1.8 CombinatorialMethods ....................................... 8 1.9 MultinomialCoefficients....................................... 13 1.10 The Probability of a Union of Events . . . . . . . . ........................ 16 1.12 Supplementary Exercises . . . . . .................................. 20

• Conditional Probability25

2.1 The Definition of Conditional Probability . . . . . ........................ 25 2.2 IndependentEvents ......................................... 28 2.3 Bayes’Theorem............................................ 34 2.4 TheGambler’sRuinProblem.................................... 40 2.5 Supplementary Exercises . . . . . .................................. 41

• Random Variables and Distributions49

3.1 RandomVariablesandDiscreteDistributions ........................... 49 3.2 ContinuousDistributions ...................................... 50 3.3 TheCumulativeDistributionFunction............................... 53 3.4 BivariateDistributions........................................ 58 3.5 MarginalDistributions........................................ 64 3.6 ConditionalDistributions ...................................... 70 3.7 MultivariateDistributions...................................... 76 3.8 FunctionsofaRandomVariable .................................. 81 3.9 FunctionsofTwoorMoreRandomVariables ........................... 85 3.10MarkovChains ............................................ 93 3.11 Supplementary Exercises . . . . . .................................. 97

• Expectation107

4.1 TheExpectationofaRandomVariable .............................. 107 4.2 PropertiesofExpectations...................................... 110 4.3 Variance................................................ 113 4.4 Moments ............................................... 115 4.5 TheMeanandtheMedian...................................... 118 Copyright © 2012 Pearson Education, Inc. Publishing as Addison-Wesley. 2 / 4

iv CONTENTS 4.6 CovarianceandCorrelation ..................................... 121 4.7 ConditionalExpectation....................................... 124 4.8 Utility . ................................................ 129 4.9 Supplementary Exercises . . . . . .................................. 134

• Special Distributions141

5.2 TheBernoulliandBinomialDistributions ............................. 141 5.3 TheHypergeometricDistributions ................................. 145 5.4 ThePoissonDistributions...................................... 149 5.5 TheNegativeBinomialDistributions................................ 155 5.6 TheNormalDistributions ...................................... 159 5.7 TheGammaDistributions...................................... 165 5.8 TheBetaDistributions........................................ 171 5.9 TheMultinomialDistributions ................................... 174 5.10TheBivariateNormalDistributions ................................ 177 5.11 Supplementary Exercises . . . . . .................................. 182

• Large Random Samples187

6.1 Introduction.............................................. 187 6.2 TheLawofLargeNumbers ..................................... 188 6.3 TheCentralLimitTheorem..................................... 194 6.4 TheCorrectionforContinuity.................................... 198 6.5 Supplementary Exercises . . . . . .................................. 199

• Estimation203

7.1 StatisticalInference ......................................... 203 7.2 PriorandPosteriorDistributions.................................. 204 7.3 ConjugatePriorDistributions.................................... 207 7.4 BayesEstimators........................................... 214 7.5 MaximumLikelihoodEstimators .................................. 217 7.6 PropertiesofMaximumLikelihoodEstimators .......................... 220 7.7 SufficientStatistics.......................................... 225 7.8 JointlySufficientStatistics...................................... 228 7.9 ImprovinganEstimator ....................................... 230 7.10 Supplementary Exercises . . . . . .................................. 234

• Sampling Distributions of Estimators 239

8.1 TheSamplingDistributionofaStatistic.............................. 239 8.2 TheChi-SquareDistributions.................................... 241 8.3 JointDistributionoftheSampleMeanandSampleVariance .................. 245 8.4 ThetDistributions.......................................... 247 8.5 Confidence Intervals ......................................... 250 8.6 BayesianAnalysisofSamplesfromaNormalDistribution .................... 254 8.7 UnbiasedEstimators......................................... 258 8.8 FisherInformation .......................................... 263 8.9 Supplementary Exercises . . . . . .................................. 267 Copyright © 2012 Pearson Education, Inc. Publishing as Addison-Wesley. 3 / 4

CONTENTS v

• Testing Hypotheses273

9.1 ProblemsofTestingHypotheses................................... 273 9.2 TestingSimpleHypotheses ..................................... 278 9.3 UniformlyMostPowerfulTests ................................... 284 9.4 Two-SidedAlternatives ....................................... 289 9.5 ThetTest............................................... 293 9.6 ComparingtheMeansofTwoNormalDistributions ....................... 296 9.7 TheFDistributions ......................................... 299 9.8 BayesTestProcedures........................................ 303 9.9 Foundational Issues . ......................................... 307 9.10 Supplementary Exercises . . . . . .................................. 309 10 Categorical Data and Nonparametric Methods 315 10.1TestsofGoodness-of-Fit....................................... 315 10.2Goodness-of-FitforCompositeHypotheses ............................ 317 10.3ContingencyTables.......................................... 320 10.4TestsofHomogeneity ........................................ 323 10.5Simpson’sParadox.......................................... 325 10.6 Kolmogorov-Smirnov Tests . . . .................................. 327 10.7RobustEstimation .......................................... 333 10.8SignandRankTests......................................... 337 10.9 Supplementary Exercises . . . . . .................................. 342 11 Linear Statistical Models 349 11.1TheMethodofLeastSquares.................................... 349 11.2Regression............................................... 353 11.3StatisticalInferenceinSimpleLinearRegression ......................... 356 11.4BayesianInferenceinSimpleLinearRegression .......................... 364 11.5TheGeneralLinearModelandMultipleRegression........................ 366 11.6AnalysisofVariance ......................................... 373 11.7TheTwo-WayLayout ........................................ 378 11.8TheTwo-WayLayoutwithReplications .............................. 383 11.9 Supplementary Exercises . . . . . .................................. 389 12 Simulation399 12.1WhatisSimulation? ......................................... 400 12.2WhyIsSimulationUseful?...................................... 400 12.3SimulatingSpecificDistributions .................................. 404 12.4ImportanceSampling......................................... 410 12.5MarkovChainMonteCarlo ..................................... 414 12.6TheBootstrap ............................................ 421 12.7 Supplementary Exercises . . . . . .................................. 425 RCodeForTwoTextExamples ..................................... 432 Copyright © 2012 Pearson Education, Inc. Publishing as Addison-Wesley.

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