Let X and Y be random variables defined on the same sample s

Let X and Y be random variables defined on the same sample space and having finite variances, let a and b be real numbers. Show that Cov(aX+bY, aX-bY) = (a^2)Var(X) - (b^2)Var(Y)

Solution

Cov(aX+bY,aX-bY)

=E((aX+bY)(aX-bY))-E(aX+bY)E(aY-bY)

=E(a^2*X^2-abXY+abXY-b^2*Y^2)- E(aX+bY)E(aY-bY)

=E(a^2*X^2-b^2*Y^2)- E(aX+bY)E(aY-bY)

=a^2Var(X)-b^2Var(Y)