Covariance, results

Summary

$X,Y,Z$ are random variables and $a,b$ are constants.

• $Cov(X,Y)=E(XY)-E(X)E(Y)$
• If $X$ and $Y$ are independent then $Cov(X,Y)=0$ . The opposite is not true.
• If $X$ and $Y$ are normally distributed and $Cov(X,Y)=0$ then $X$ and $Y$ are independent
• $Cov(X,X)=Var(X)$
• $Cov(X,a)=0$
• $Cov\left( X,Y \right)=Cov(Y,X)$
• $Cov(X+Y,Z)=Cov(X,Z)+Cov(Y,Z)$
• $Cov(aX,bY)=abCov(X,Y)$
• $Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y)$