Lecture11CovarianceandcorrelationTheSampleMean
CovarianceandCorrelationCovariance:measuretheassociationbetweentworandomvariables. LetXandYberandomvariableshavingaspecifiedjointdistribution,andletE(X)=,E(Y)=, Var(X)=,Var(Y)=.ThecovarianceofXandY,isdefinedasIf,,thenCov(X,Y)willbefinite.Cov(X,Y)canbepositive,negative,orzero.
CorrelationIf,,thecorrelationofXandY,isdefinedas Therangeofpossiblevaluesofthecorrelationis:
Theorem(Schwarzinequality):ForanyrandomvariablesUandV, Proof.(a)If,thenPr(U=0)=1.ItfollowsthatPr(UV=0)=1.SoE(UV)=0andtherelationissatisfied.If,therelationisalsosatisfied. (b)Ifeitherorisinfinite,apparentlytherelationissatisfied.
(c)If,
Let,thenXandYarepositivelycorrelated: XandYarenegativelycorrelated: XandYareuncorrelated:
PropertiesofCovarianceandCorrelation Theorem.ForanyrandomvariablesXandYsuchthatand, Cov(X,Y)=E(XY)-E(X)E(Y) Proof.
Theorem.IfXandYareindependentrandomvariableswithand ,then Proof.IfXandYareindependent,thenE(XY)=E(X)E(Y).Therefore, Cov(X,Y)=E(XY)-E(X)E(Y)=0. Itfollowsthat
Remark:Twouncorrelatedrandomvariablescanbedependent. Example4.6.3.SupposethatXcantakeonlythreevalues–1,0,and1andeachofthesethreevalueshasthesameprobability.LetYbedefinedby.WeshallshowthatXandYaredependentbutuncorrelated. Proof.ApparentlyXandYaredependent. XandYareuncorrelated.
Theorem.SupposeXisarandomvariablewith,supposethatY=aX+bwhere.Ifa0,then