Consider random variables X and Y that are jointly distributed based on the joint probability distribution function f(x, y) with 0 f(x, y) 1 and P xX P yY f(x, y) = 1 for any x X and any y Y . Using the definitions for expected values and co-variances of of jointly-distributed random variables show that:

(a) Cov(aX, bY ) = abCov(X, Y )

(b) Cov(X, Y + Z) = Cov(X, Y ) + Cov(X, Z)

(c) V ar(aX + bY ) = a2V ar(X) + b 2V ar(Y ) + 2abCov(X, Y )