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1、4.ContinuousRandomVariable连续型随机变量Continuousrandomvariablesappearwhenwedealwillquantitiesthataremeasuredonacontinuousscale.Forinstance,whenwemeasurethespeedofacar,theamountofalcoholinaperson'sblood,thetensilestrengthofnewalloy.Weshalllearnhowtodetermineandworkwithprobabilitiesrelatingtocontinuousran
2、domvariablesinthischapter.Weshallintroducetotheconceptoftheprobabilitydensityfunction.4.1ContinuousRandomVariable1.DefinitionDefinition4.1.1Afunctionf(x)definedoniscalledaprobabilitydensityfunction（概率密度函数）if:(i);(ii)f(x)isintergrable(可积的)onand.Definition4.1.2Letf(x)beaprobabilitydensityfunction.IfX
3、isarandomvariablehavingdistributionfunction,(4.1.1)thenXiscalledacontinuousrandomvariablehavingdensityfunctionf(x).Inthiscase,.(4.1.2)2.几何意义643.NoteInmostapplications,f(x)iseithercontinuousorpiecewisecontinuoushavingatmostfinitelymanydiscontinuities.Note1ForarandomvariableX,wehaveadistributionfunct
4、ion.IfXisdiscrete,ithasaprobabilitydistribution.IfXiscontinuous,ithasaprobabilitydensityfunction.Note2LetXbeacontinuousrandomvariable,thenforanyrealnumberx,.4.ExampleExample4.1.2Findksothatthefollowingcanserveastheprobabilitydensityofacontinuousrandomvariable:SolutionTosatisfytheconditions(4.1.1),k
5、mustbenonnegative,andtosatisfythecondition(4.1.2)wemusthavesothat.(Cauchydistribution柯西分布)□Example4.1.3CalculatingprobabilitiesfromtheprobabilitydensityfunctionIfarandomvariablehastheprobabilitydensity64Findtheprobabilityfromthatitwilltakeonvalue(a)between0and2;(b)greaterthan1.SolutionEvaluatingthe
6、necessaryintegrals,weget(a)(b)Example4.1.4DeterminingthedistributionfunctionofX,itisknownSolutionPerformingthenecessaryintegrations,weget□5.meanDefinition4.1.2LetXbeacontinuousrandomvariablehavingprobabilitydensityfunctionf(x).Thenthemean(orexpectation)ofXisdefinedby,(4.1.3)providedtheintegralconve
7、rgesabsolutely.Iftheintegral(4.1.3)doesnotconvergesabsolutely（绝对收敛）,wesaythemeanofXdoesnotexist.Themeanofcontinuousrandomvariablehasthesimilarpropertiesasdiscreterandomvariable.Ifg(X)isanintegrablefunctiono
