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
• \(Cov(X,X)=Var(X)\)
• \(Cov(X,a)=0\)
• \(Cov\left( X,Y \right)=Cov(Y,X)\)
• \(Cov(X+Y,Z)=Cov(X,Z)+Cov(Y,Z)\)
• \(Cov(aX,bY)=abCov(X,Y)\)
• \(Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y)\)