Covariance

Definition

• For RV $X$ with Mean ${\mu }_x=E(X)$ and Variance ${\sigma }_X^2=V(X)$
• For RV $Y$ with Mean ${\mu }_Y=E(Y)$ and Variance ${\sigma }_Y^2=V(Y)$
• The covariance of $X$ and $Y$ defined as $Cov(X,Y)=E[(X-{\mu }_X)(Y-{\mu }_Y)]$

Properties

• $Cov(aX,bY)=abCov(X,Y)$
• $Cov(X,X)=V(X)$
• $Cov(X,Y)=E(XY)-{\mu }_X{\mu }_Y$
• If $X\perp Y⟹Cov(X,Y)=0$
• Variance of linear combinations: $V(X+Y)=V(X)+V(Y)+2Cov(X,Y)$
• Linearity of covariances $Cov({\Sigma }_{i=1}^n{a}_i{X}_i,{\Sigma }_{j=1}^m{b}_j{Y}_j)={\Sigma }_{i=1}^n{\Sigma }_{j=1}^m{a}_i{b}_{j}Cov(X_i,Y_j)$