003 Statistical Properties英文学习材料

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1、StatisticalPropertiesRichardYiDaXuSchoolofComputing&Communication,UTSMarch16,2015RichardYiDaXuStatisticalPropertiesChangeinvariablesLetY=g(X):FY(y)=P(g(X)y)=P(Xg1(y))=F(g1(y))X@FY(y)FX(g1(y))@xFY(y)==@y@x@y1@g1(y)=fx(g(y))@y1@g1(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(jXjk)k2substitute

4、X!(X)2andb!2k2:222E[(X)2]1=)Pr(X)k=2k2k21=)Pr(jXjk)k2RichardYiDaXuStatisticalPropertiesChebyshev’sinequalityapplications(1)Providesboundsofrandomvariablesfromanydistributionswhentheirmeansandvariancesareknown.Eachktellsusonebound,forexample,whenk=2:11Pr(jXj2)=)Pr(X

5、2;+X2)441=)Pr(X2;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(jXj>k)k221=)Pr(jXn1j>)=!0asn!12n

6、2PDefinitionXnconvergesinprobabilitytotherandomvariableXi.e.,Xn!X:Pr(jXnXj>)!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(jXj>)=)PrjSnj>2n2=)PrjSnj>!0asn!1PIThelawoflargenumbersstatesthatSn!Pr(jSnj>)!0asn!1RichardYiDaXuStatisticalPropertiesUniqueness