Covariance and correlation - Main Concepts

Covariance and correlationLet random variablesX,Ywith means X, Yrespectively. The Covariance , denoted withcov(X,Y), is ameasure of the association :cov(X,Y) =E(X X)(Y Y)This can be simplified as follows:cov(X,Y) =E(X X)(Y Y) =E(XY) YE(X) XE(Y) + X YTherefore,cov(X,Y) =E(XY) (EX)(EY)Note: IfX,Yare independent thenE(XY) = (EX)E(Y) Thereforecov(X,Y) = ,X,Y,Zrandom variables, anda,b,c,dconstants: Findcov(a+X,Y)cov(a+X,Y) =E(a+X a+X)(Y Y) =E(a+X X a)(Y Y)Therefore,cov(a+X,Y) =cov(X,Y). Findcov(aX,bY)cov(aX,bY) =E(aX aX)(bY bY) =E(aX a X)(bY b Y)Therefore,cov(aX,bY) =abE(X X)(Y Y) =ab cov(X,Y) Findcov(X,Y+Z)cov(X,Y+Z) =E(X X)(Y Y+Z) =E(X X)(Y+Z Y Z)Orcov(X,Y+Z) =E(X X)(Y Y+Z Z) =E(X X)(Y Y) +E(X X)(Z Z)Therefore,cov(X,Y+Z) =cov(X,Y) +cov(X,Z) Using the results above we can findcov(aW+bX,cY+dZ).

Correlation: However, the covariance depends on the scale of measurement and so it is not easy to say whether a particular covariance is small or large. The problem is solved by standardize the value of covariance (divide it by ˙ X˙ Y), to get the so called coe cient of correlation ˆ XY. ˆ= cov(X;Y) ˙ X˙ Y; Always, 1 ˆ 1 cov(X;Y) = ˆ ...

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  Correlations, Covariance, Covariance and correlation

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