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).cov(aW+bX,cY+dZ) =ab cov(W,Y) +ad cov(W,Z) +bc cov(X,Y) +bd cov(X,Z) correlation :However, the Covariance depends on the scale of measurement and so it is not easy to say whether aparticular Covariance is small or large.

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˙ X˙ Y

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  Correlations, Covariance, Covariance and correlation

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