[理学]概率论与数理统计英文第三章

《[理学]概率论与数理统计英文第三章》由会员上传分享，免费在线阅读，更多相关内容在应用文档-天天文库。

1、3.RandomVariables3.1DefinitionofRandomVariablesInengineeringorscientificproblems,wearenotonlyinterestedintheprobabilityofevents,butalsointerestedinsomevariablesdependingonsamplepoints.（定义在样本点上的变量）Forexample,wemaybeinterestedinthelifeofbulbsproducedbyacertaincompany,ortheweightofcowsinace

2、rtainfarm,etc.Theseideasleadtothedefinitionofrandomvariables.1.randomvariabledefinitionDefinition3.1.1Arandomvariableisarealvaluedfunctiondefinedonasamplespace;i.e.itassignsarealnumbertoeachsamplepointinthesamplespace.Herearesomeexamples.Example3.1.1Afairdieistossed.Thenumbershownisarand

3、omvariable,ittakesvaluesintheset.Example3.1.2ThelifeofabulbselectedatrandomfrombulbsproducedbycompanyAisarandomvariable,ittakesvaluesintheinterval.Sincetheoutcomesofarandomexperimentcannotbepredictedinadvance,theexactvalueofarandomvariablecannotbepredictedbeforetheexperiment,wecanonlydis

4、cusstheprobabilitythatittakessome52valueorthevaluesinsomesubsetofR.2.DistributionfunctionDefinition3.1.2Letbearandomvariableonthesamplespace.Thenthefunction.iscalledthedistributionfunctionofNoteThedistributionfunctionisdefinedonrealnumbers,notonsamplespace.Example3.1.3Letbethenumberweget

5、fromtossingafairdie.Thenthedistributionfunctionofis(Figure3.1.1)Figure3.1.1ThedistributionfunctioninExample3.1.33.PropertiesThedistributionfunctionofarandomvariablehasthefollowingproperties:(1)isnon-decreasing.52Infact,if,thentheeventisasubsetoftheevent,thus(2),.(3)Forany,.Thisistosay,th

6、edistributionfunctionofarandomvariableisrightcontinuous.Example3.1.4LetbethelifeofautomotivepartsproducedbycompanyA,assumethedistributionfunctionofis(inhours)Find,.SolutionBydefinition,.Question：Whataretheprobabilitiesand?Example3.1.5Aplayertossestwofairdice,ifthetotalnumbershownis6ormor

7、e,theplayerwins$1,otherwiseloses$1.Letbetheamountwon,findthedistributionfunctionof.Solution52Letbethetotalnumbershown,thentheeventscontainssamplepoints,.Thus,AndsoThusFigure3.1.2ThedistributionfunctioninExample3.1.5Thedistributionfunctionofrandomvariablesisaconnectionbetw