Multiple_Random_Variables

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1、NOTESONMULTIPLERANDOMVARIABLES1.1DISTRIBUTIONANDDENSITYFUNCTIONSTheconsiderationofmorethantworandomvariablesisbyandlargejusttheoftennaturalextensionsofthenotionsandresultsassociatedwiththeconsiderationoftworandomvariables.Thesefewpagesbrieflyrestatethemainpointsint

2、hegeneralcase.GivenafinitecollectionofnrandomvariablesX1,X2,...,Xn,theirjointstatisticalpropertiesarefullydeterminedbyspecifyingtheirjointdistributionfunctiondefinedbyFX1X2···Xn(x1,x2,...,xn)=P(X1x1,X2x2,...,Xnxn).(1.1)Thefivepropertiesthatweregivenearlierforthejo

3、intdistributionfunctionoftworandomvariablesapply(intheanalogousformsfornrandomvariables)tothegeneralsituation:Property1.1.FX1X2···Xn(1,1,...,1),limFX1X2···Xn(x1,x2,...,xn)=1.(1.2)xi!1alliProperty1.2.FX1X2···Xn(x1,x2,...,xm1,1,xm+1,xm+2,...,xn),limFX1X2···Xn(x1,x

4、2,...,xn)=0xm!1foranyrealnumbersx1,x2,...,xm1,xm+1,xm+2,...,xn,wheremisanyintegerfrom1ton.Property1.3.Foranyrealnumbersx1,x2,x3,...,xnandy1,y2,y3,...,ynsuchthatxiyiforalli2{1,2,...,n},FX1X2···Xn(x1,x2,...,xn)FX1X2···Xn(y1,y2,...,yn).(1.3)Property1.4.Foranyreal

5、numbersx1,x2,x3,...,xnandy1,y2,y3,...,ynsuchthatxiyiforalli2{1,2,...,n},P(x1

6、Xn(x1,x2,y3,...,yn)+FX1X2···Xn(x1,y2,x3,...,yn)+···+FX1X2···Xn(x1,y2,...,xn1,yn)+FX1X2···Xn(x1,y2,...,yn1,xn)+FX1X2···Xn(y1,x2,x3,y4...,yn)+FX1X2···Xn(y1,x2,y3,x4,...,yn)+···+FX1X2···Xn(y1,x2,...,xn1,yn)+FX1X2···Xn(y1,x2,y3,...,yn1,xn)+···+FX1X2···Xn(y1,y2,...

7、,yn2,xn1,xn)}{FX1X2···Xn(x1,x2,x3,y4,...,yn1,yn)+···+FX1X2···Xn(y1,y2,y3,...,xn2,xn1,xn)}+···+(1)n{F(x,x,...,x,x)}0.(1.4)X1X2···Xn12n1n1ELG3126RANDOMSIGNALSANDSYSTEMSWinter2013Property1.5.limFX1X2···Xn(x1+✏1,x2+✏2,...,xn+✏n)=FX1X2···Xn(x1,x2,...,xn).(1.5)

8、+✏i!0alli2{1,2,3,...,n}Analogoustotheresultfortworandomvariables,wefindthatitisalwayspossibletofindrandomvariablesX1,X2,...,Xnwhosedistributionfunctionwil