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时间:2019-08-30
《003 Statistical Properties英文学习材料》由会员上传分享,免费在线阅读,更多相关内容在教育资源-天天文库。
1、StatisticalPropertiesRichardYiDaXuSchoolofComputing&Communication,UTSMarch16,2015RichardYiDaXuStatisticalPropertiesChangeinvariablesLetY=g(X):FY(y)=P(g(X)y)=P(Xg 1(y))=F(g 1(y))X@FY(y)FX(g 1(y))@xFY(y)==@y@x@y 1@g 1(y)=fx(g(y))@y 1@g 1(y)=fx(g(y))@yRichardYiDaXuStatisticalPropertiesUsef
2、ulinequalities:Markov’sinequalityMarkovsinequalityLetXbeanonnegativerandomvariable.Then,foranyb2R+:E[X]Pr(Xb)bWhy?ZZZ1b1E[X]=xp(x)dx=xp(x)dx+xp(x)dx00bZ1=)E[X]xp(x)dxbZ1bp(x)dxbZ1=bp(x)dxb=bPr(Xb)howisthisuseful?providesanupperboundofprobabilitythatanonnegativerandomvariableisgreatert
3、hananarbitrarypositiveconstantbyrelatingaprobabilitytoanexpectation.RichardYiDaXuStatisticalPropertiesUsefulinequalities:Chebyshev’sinequalityLetXbeanonnegativerandomvariable.Then,foranyb2R+:E[X]Pr(Xb)bsubstituteX!(X )2andb!k2:22E[(X )2]2=)Pr(X )k=k2k22=)Pr(jX jk)k2substitute
4、X!(X )2andb!2k2:222E[(X )2]1=)Pr(X )k=2k2k21=)Pr(jX jk)k2RichardYiDaXuStatisticalPropertiesChebyshev’sinequalityapplications(1)Providesboundsofrandomvariablesfromanydistributionswhentheirmeansandvariancesareknown.Eachktellsusonebound,forexample,whenk=2:11Pr(jX j2)=)Pr( X
5、2;+X2)441=)Pr(X 2;X+2)413=)Pr( 2X+2)1 =44ForGuassiandistribution,Pr( 2X+2)0:995RichardYiDaXuStatisticalPropertiesChebyshev’sinequalityapplications(2)LetX1n2Gamma(n;),therefore:n2111E[Xn]=n=1VAR[Xn]=n=nnnTherefore,2Pr(jX j>k)k221=)Pr(jXn 1j>)=!0asn!12n
6、2PDefinitionXnconvergesinprobabilitytotherandomvariableXi.e.,Xn !X:Pr(jXn Xj>)!0asn!1RichardYiDaXuStatisticalPropertiesChebyshev’sinequalityapplications(3)lawoflargenumbersILetX1;X2;:::Xnbeasequenceofi.i.d.randomvariableswithmeanandfinitevariance2ILetSn=X1+X2++Xn.nVAR(aX+bY)=a2VAR(X)+b
7、2VAR(Y)ifXandYareindependantE[X1]E[Xn]E[Sn]=++=nnVAR[X1]VAR[Xn]2VAR[Sn]=++=n2n2nTherefore,VAR[Sn]2Pr(jX j>)=)PrjSn j>2n2=)PrjSn j>!0asn!1PIThelawoflargenumbersstatesthatSn !Pr(jSn j>)!0asn!1RichardYiDaXuStatisticalPropertiesUniqueness
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