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1、3.RandomVariables3.1DefinitionofRandomVariablesInengineeringorscientificproblems,wearenotonlyinterestedintheprobabilityofevents,butalsointerestedinsomevariablesdependingonsamplepoints.(定义在样本点上的变量)Forexample,wemaybeinterestedinthelifeofbulbsproducedbyacertaincompany,ortheweightofcowsinacertainfarm
2、,etc.Theseideasleadtothedefinitionofrandomvariables.1.randomvariabledefinitionDefinition3.1.1Arandomvariableisarealvaluedfunctiondefinedonasamplespace;i.e.itassignsarealnumbertoeachsamplepointinthesamplespace.Herearesomeexamples.Example3.1.1Afairdieistossed.Thenumbershownisarandomvariable,ittakes
3、valuesintheset.Example3.1.2ThelifeofabulbselectedatrandomfrombulbsproducedbycompanyAisarandomvariable,ittakesvaluesintheinterval.Sincetheoutcomesofarandomexperimentcannotbepredictedinadvance,theexactvalueofarandomvariablecannotbepredictedbeforetheexperiment,wecanonlydiscusstheprobabilitythatittak
4、essome52valueorthevaluesinsomesubsetofR.2.DistributionfunctionDefinition3.1.2Letbearandomvariableonthesamplespace.Thenthefunction.iscalledthedistributionfunctionofNoteThedistributionfunctionisdefinedonrealnumbers,notonsamplespace.Example3.1.3Letbethenumberwegetfromtossingafairdie.Thenthedistribut
5、ionfunctionofis(Figure3.1.1)Figure3.1.1ThedistributionfunctioninExample3.1.33.PropertiesThedistributionfunctionofarandomvariablehasthefollowingproperties:(1)isnon-decreasing.52Infact,if,thentheeventisasubsetoftheevent,thus(2),.(3)Forany,.Thisistosay,thedistributionfunctionofarandomvariableisright
6、continuous.Example3.1.4LetbethelifeofautomotivepartsproducedbycompanyA,assumethedistributionfunctionofis(inhours)Find,.SolutionBydefinition,.Question:Whataretheprobabilitiesand?Example3.1.5Aplayertossestwofairdice,ifthetotalnumbershownis6ormore,theplayerwins$1,otherwiseloses$1.Letbetheamountwon,f
7、indthedistributionfunctionof.Solution52Letbethetotalnumbershown,thentheeventscontainssamplepoints,.Thus,AndsoThusFigure3.1.2ThedistributionfunctioninExample3.1.5Thedistributionfunctionofrandomvariablesisaconnectionbetw