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1、3.RandomVariables3.1DefinitionofRandomVariablesInengineeringorscientificproblems,wearenotonlyinterestedintheprobabilityofevents,butalsointerestedinsomevariablesdependingonsamplepoints.（定义在样本点上的变量）Forexample,wemaybeinterestedinthelifeofbulbsproducedbyacertaincompany,ortheweightofcowsi
2、nacertainfarm,etc.Theseideasleadtothedefinitionofrandomvariables.1.randomvariabledefinitionDefinition3.1.1Arandomvariableisarealvaluedfunctiondefinedonasamplespace;i.e.itassignsarealnumbertoeachsamplepointinthesamplespace.Herearesomeexamples.Example3.1.1Afairdieistossed.Thenumbershow
3、nisarandomvariable,ittakesvaluesintheset.Example3.1.2ThelifeofabulbselectedatrandomfrombulbsproducedbycompanyAisarandomvariable,ittakesvaluesintheinterval.Sincetheoutcomesofarandomexperimentcannotbepredictedinadvance,theexactvalueofarandomvariablecannotbepredictedbeforetheexperiment,
4、wecanonlydiscusstheprobabilitythatittakessomevalueorthevaluesinsomesubsetofR.2.DistributionfunctionDefinition3.1.2Letbearandomvariableonthesamplespace.Thenthefunction.iscalledthedistributionfunctionofNoteThedistributionfunctionisdefinedonrealnumbers,notonsamplespace.Example3.1.3Letbe
5、thenumberwegetfromtossingafairdie.Thenthedistributionfunctionofis(Figure3.1.1)Figure3.1.1ThedistributionfunctioninExample3.1.33.PropertiesThedistributionfunctionofarandomvariablehasthefollowingproperties:(1)isnon-decreasing.Infact,if,thentheeventisasubsetoftheevent,thus(2),.(3)Forany
6、,.Thisistosay,thedistributionfunctionofarandomvariableisrightcontinuous.Example3.1.4LetbethelifeofautomotivepartsproducedbycompanyA,assumethedistributionfunctionofis(inhours)Find,.SolutionBydefinition,.Question：Whataretheprobabilitiesand?Example3.1.5Aplayertossestwofairdice,ifthetota
7、lnumbershownis6ormore,theplayerwins$1,otherwiseloses$1.Letbetheamountwon,findthedistributionfunctionof.SolutionLetbethetotalnumbershown,thentheeventscontainssamplepoints,.Thus,AndsoThusFigure3.1.2ThedistributionfunctioninExample3.1.5Thedistributionfunctionofrandomvariablesisaconnecti
8、onbetweenpro
