Correlation Coefficient and Covariance of Two Equations

Question

Can anyone help me find the covariance and correlation coefficient?

Thanks in advance.

Answer 1

Comment. For independent $X$ and $Y,$ key equations are:

Expectation: $E(aX + bY) = aE(X)+bE(Y),$

Variance: $Var(aX + bY) = a^2Var(X) + b^2Var(Y),$

Covariance: $$Cov(aX + bY, cX + dY) = ac\,Cov(X,X) + ad\,Cov(X,Y) + bc\,Cov(X,Y) + bd\,Cov(Y,Y)\\ = ac\,Var(X) + ad\,Cov(X,Y) + bc\,Cov(X,Y) + bd\,Var(Y).$$

You should be able to find similar equations in your text or notes. Once you have covariance and variances, you can find correlation: $Cor(X,Y) = \frac{Cov(X,Y)}{SD(X)\,SD(Y)}.$