样本方差无偏性证明.pdf

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1、TheDerivationofSampleVariance'sunbiasednessAnarticleinLATEX2eMaiarJanuary23,2013Contents1Question11.1Variance................................11.2SampleVariance...........................11.3Unbiasedness.............................22Answer42.1Derivation......................

2、.........42.2Tips..................................5Chapter1Question1.1VarianceWell,andherebeginsmyrstarticle(actuallyabook,that'snotajoke...)1writteninLATEX2e.It'sdenedasfollows:pi=P(X=xi)(1.1)XnE(X)=xipi(1.2)i=12Xn2D(X)=EXE(X)=xiE(X)pi(1.3)i=11.2SampleVarianceWei

3、ntroduceanewvariableXasequation(1.4)onpage1shows:Xn1X=Xi(1.4)ni=1Then1Xn1XnE(X)=EXi=E(Xi)=(1.5)nni=1i=1XnXn211D(X)=DXi=D(Xi)=(1.6)n2n2ni=1i=11ù´·1˜‡5"2QuestionSampleVarianceisdenedasequation(1.7):Xn212S=(XiX)(1.7)n1i=11.3UnbiasednessTheunbiasednessofSampleMeanandSa

4、mpleVarianceisasfollows:8

5、i2222E(X)=D(X)+=+n2X1Xi+X2Xi+


+Xi+


+XnXiE(XXi)=Enn1X=E(XX)+E(X2)kiink=0k6=i1222=((n1)+(+))n2=+2n2=E(X)2.2Tips5Nowlet'sbeginthederivation:n1XE(S2)=E(XX)2in1i=1Xn122=EXi2XXi+Xn1i=1Xn122=EXi2XXi+Xn1i=1Xn122=2n+(n+1)2E(XXi)n1i=121222=2

6、n+(n+1)2n(+)n1n=2=D(X)That'sall.Hopeyouenjoyit.2.2TipsWhat'sthedierencebetweenxiandXi??Thinkautit,whenyougetananswer,youcangobacktoclicktheblankpage3andcommentonmyarticle...andhereitends.