概率论与数理统计英文版3

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

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

3、henumberXshownisarandomvariable,ittakesvaluesintheset{1,2,"6}.,Example3.1.2ThelifetofabulbselectedatrandomfrombulbsproducedbycompanyAisarandomvariable,ittakesvaluesintheinterval(0,)∞.,Sincetheoutcomesofarandomexperimentcannotbepredictedinadvance,theexactvalueofarandomvariableca

4、nnotbepredictedbeforetheexperiment,wecanonlydiscusstheprobabilitythatittakessome27valueorthevaluesinsomesubsetofR.2.DistributionfunctionDefinition3.1.2LetXbearandomvariableonthesamplespaceS.ThenthefunctionFX()(=≤PXx).x∈RiscalledthedistributionfunctionofXNoteThedistributionfunct

5、ionFX()isdefinedonrealnumbers,notonsamplespace.Example3.1.3LetXbethenumberwegetfromtossingafairdie.ThenthedistributionfunctionofXis(Figure3.1.1)⎧0,ifx<1;⎪⎪nFx()=≤⎨,ifnxn<+1,n=1,2,,5;"⎪6⎪⎩1,ifx≥6.Figure3.1.1ThedistributionfunctioninExample3.1.33.PropertiesThedistributionfunction

6、Fx()ofarandomvariableXhasthefollowingproperties:(1)Fx()isnon-decreasing.28Infact,ifx≤x,thentheevent{}X

7、nctionFx()ofarandomvariableXisrightcontinuous.Example3.1.4LetXbethelifeofautomotivepartsproducedbycompanyA,assumethedistributionfunctionofXis(inhours)x⎧−⎪1,0−≥ex2000;FxPXx()=≤()=⎨⎩⎪0,x<0.FindPX(≤2000),PX(1000<≤3000).SolutionBydefinition,−1PX(≤=2000)F(2000)1=−e=0.6321.PX(1000<≤3

8、000)=P(X≤3000)−P(X≤1000)−−1.50.5=−=FF(3000)(1000)(1−ee