Transcription of Covariance and Correlation Math 217 Probability and ...
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Covariance and Correlation Math 217 Probability and ...
Covariance and CorrelationMath 217 Probability and StatisticsProf. D. Joyce, Fall joint random vari-ables. TheircovarianceCov(X,Y) is defined byCov(X,Y) =E((X X)(Y Y)).Notice that the variance ofXis just the covarianceofXwith itselfVar(X) =E((X X)2) = Cov(X,X)Analogous to the identity for varianceVar(X) =E(X2) 2 Xthere is an identity for covarianceCov(X) =E(XY) X YHere s the proof:Cov(X,Y)=E((X X)(Y Y))=E(XY XY X Y+ X Y)=E(XY) XE(Y) E(X) Y+ X Y=E(XY) X YCovariance can be positive, zero, or indicates that there s an overall tendencythat when one variable increases, so doe the other,while negative indicates an overall tendency thatwhen one increases the other independent variables, then theircovariance is 0:Cov(X,Y) =E(XY) X Y=E(X)E(Y) X Y= 0 The converse, however, isnotalways (X,Y) can be 0 for variables that are not an example where the Covariance is 0 butXandYaren t independent, let there be threeoutcomes, ( 1,1), (0, 2), and (1,1), all with thesame probability13.
dard deviations, the correlation becomes bounded ... kind of thing that goes on in linear algebra. In fact, it is the same thing exactly. Take a set of real-valued random variables, not necessarily inde-pendent. Their linear combinations form a vector space. Their covariance is …
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