Question

4. [-14 Points] DETAILS (4pt) The variance of random variable X is 1 and the variance of random variable Y is 4. The correlation coefficient between the two random variables X and Y is 0.2. (a) (1pt) Find the covariance between X and Y. (b) A new random variable Z is given by Z = 2X + 1. Find the covariance between X and Z. (1pt) Find the covariance between Y and Z. (2pt)

Answer 1

(6) Now E (2) - E (2x+1) E(Z) = 2 E(X) + 1 and Ecxz) = E(X(2x H) E (2x2+x) - 2E (x²)+ E(X) Now E CXZ) - ECX E(Z) Corc, Z) = 2 ECKY + E CX) - E(X) (2EWT+1) = 2 ECX) + E \ ) – 2CE WIJ? EK ) = 2(Eck") (Exo 3² ) 2 ( var(x)) - 2x 2 covex,Z) = 2 T.

Now ELYZ) = E CYC 2x+1) - E (2xy + y) = 2E OXY) + E(Y) Core Y,Z) = ELYZ) - ELY) ELE) 2 ECXY+ELY) – ECV) (2EX+1) = 2 Ecxy) + EAS-2E(X) ECO) - Ed = 2 (ECXY) – E XXI ECY)) = 2 COUCX, Y). 2x0.4 covcy, z) = 0.8 T