Question

A fair die is rolled 2 times, and the results are Z1 and Z2. Define X = Z1+Z2 and Y = Z1−Z2. (a) Show that Cov(X, Y ) = 0. (b) Prove that X and Y are not independent. (Hint: find a pair of numbers k and l for which P(X = k, Y = l) 6= P(X = k) × P(Y = l).)

Answer 1

(a).

E(XY) = 0 = E(X) * E(Y) , thus covariance is 0.

(b). P(X=2,Y=-1) = 0

and P(X=2)*P(Y=-1) 0. Thus X and Y are not independent.