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1、TheUniformandNormalDistributionWhenarandomvariableXcantakeonanyvalueinaninterval,wecallitacontinuousrandomvariableanddescribeitsoutcomesusingaprobabilitydensityfunction.InsteadoftalkingabouttheprobabilitythatXtakesonaspecificvalueP(X=x)aswedointhediscretecase
2、,wetalkabouttheprobabilitythatXfallsinanintervalsuchas[c,d]andwriteitas.TheprobabilitythatXtakesonaspecificvalueinthecontinuouscaseis0.TheprobabilitythatXfallsinaninterval[c,d]isdeterminedbycalculatingtheareaunderthedensityfunctionthatisboundedbyc,d,andthex-a
3、xis.Inthecontinuouscase,probabilitiescorrespondtoareas.UniformDistributionTheuniformdistributionispossiblythesimplestexampleofacontinuousrandomvariable.Itsprobabilitydensityfunction(“density”forshort)isgivenbyThegraphofthisdensityisshownbelow:UniformDensityab
4、Noticethattheheightoftheelevatedhorizontallineispreciselysothatthetotalareaunderthecurveis1.Themeanofauniformrandomvariableis.Thevarianceis.Example.SupposeUfollowsauniformdistributionon[3,8](usuallywrittenU[3,8]).(a)WhatistheprobabilitythatUfallsbetween4and6?
5、(b)WhatistheprobabilitythatUfallsbetween0and5?38(a)AREA1/56421CopyrightJohnSemple2007TheNormalDistributionTheNormaldistributionisthemostimportantcontinuousdistributioninallofstatistics.The“bellcurve,”asitisfrequentlyreferredto,hasadensityfunctiongivenbyfor,wh
6、ereandareparameters()whichaffecttheshapeofthedistribution.ThechoiceofGreeklettersisprophetic.IfXisarandomvariablewhosedensityfunctionis,thenXhasanormaldistributionwithmeanandvariance(equivalently,thestandarddeviationofXis).Notation:AnormalrandomvariableXwithm
7、eanandvarianceisdescribedusingtheshorthandnotation.Thetilde(~)isread“isdistributedas.”CalculationswiththeNormalDistributioninExcelExample.SupposeaBookRepinformsyouthathis/herpublishercanprovideyouwithahighqualitycustomizedstatisticstextbookforyourMBAstudents,
8、whojusthappentobeattendinganelite,privateBusinessSchoolinNorthTexaswhosenamerhymeswiththeword“Sox.”However,astheRepacknowledges,thistypicallyrequires6-8weeksofproductiontimeafteritisorder
