multivariate probability distributions

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1、Chapter5MultivariateProbabilityDistributionsPhysicianshavenothingtodowithwhatiscalledthelawoflargenumbers,alawwhich,accordingtoagreatmathematician’sexpressionisalwaystrueingeneralandfalseinparticular.ClaudeBernard,Frenchphysiologist(1813-1878)5.1OrientationWefrequentlyhavetheoccasiontostu

2、dythemutualinteractionsbetweentwoormorevariables.Howusefulareastudent’scollegegradepointaverage(GPA)andscoreonthelawschooladmissionstest(LSAT)aspredictorvariablesforastudent’sfutureperformanceinlawschool?Howdoesdiversificationofaninvestmentportfolioreducerisk,andhowdoesitaffectthereturns?Fo

3、rexample,whenoilpricessoarsodothestockpricesofoilcompanies,whilethoseofautomobilemanufacturersandairlinesfall.Modernportfoliotheorybeginswithananalysisofthejointvariationofthedifferentassetclasses(suchasstocks,bonds,andmutualfunds)intheportfolio.Inenvironmentalstudiestheairqualityofametrop

4、olitanregionisdeterminedbysimultaneouslymeasuringseveralpollutantssuchasozone,nitrogenoxide,hydrocarbons,etc.Beforeundertakingaplantoreducethepollutionlevelstheenvironmentalscientistmuststudytheirinteractionssincereducingthelevelofjustonepollutantcouldactuallyincreasethecombinedharmfuleffe

5、ctsoftheothers.Statisticiansusethejointdistributionfunctiontostudythejointvariationsamongthevariables.Webeginwiththediscretecaseandfollowwiththecontinuouscase,theonlydifferencebetweenthemisthat,inthelattercase,weusesomebasictoolsfrommultivariatecalculus.OrganizationofChapter1.Section5.2:Th

6、eJointDistributionFunction:DiscreteRandomVariables2.Section5.2.1:IndependentRandomVariables3.Section5.3:TheMultinomialDistribution4.Section5.4:MeanandVarianceofaSumofRandomVariables5.Section5.4.1:TheLawofLargeNumbersforSumsofIndependentandIdenticallyDistributed(iid)RandomVariables6.Sectio

7、n5.4.2:TheCentralLimitTheorem7.Section5.5:WhyStockPriceshaveaLognormalDistribution:AnApplicationoftheCentralLimitTheorem205206IntroductiontoProbabilityandStatisticsforScience,Engineering,andFinance8.Section5.5.1:TheBinomialLatticeModelasanApproximationtoaContinuousT