Question

4. Recall that the covariance of random variables X, and Y is defined by Cov(X,Y) = E(X - Ex)(Y - EY) (a) (2pt) TRUE or FALSE (circle one). E(XY) 0 implies Cov(X, Y) = 0. (b) (4 pt) a, b, c, d are constants. Mark each correct statement ( ) Cov(aX, cY) = ac Cov(X, Y) ( ) Cor(aX + b, cY + d) = ac Cov(X, Y) + bc Cov(X, Y) + da Cov(X, Y) + bd ( ) Cov(X + b, x + d) = Var(X) ( ) Cov(X, c) =0 (c) (6 pt) From the list below, select conditions on the random variables Xi,..., Xn Σ:.ix.)-Σ.ivar(Χ.). such that () Xi, Xj are independent for each i j. ) For each i, X is independent of the collection of r.v. {Xj)ii ()EXiXj = 0 for each ij ( ) EXX, = EX,EX, for each i j ( ) Cov(Xi, Xs) = 0 for each i j ()EX/ = 0 for each i in general the following is TRUE: Var(

Answer 1

a)

Cov(X, Y) = E[XY] - E[X] E[Y]

if E[XY] = 0, then Cov(X, Y) = - E[X] E[Y]

Thus, if E[X], E[Y] $\ne$ 0 , then Cov(X, Y) $\ne$ 0

Thus the statement is False.

b)

Cov(aX, cY) = ac Cov(X, Y) - Correct Statement

Cov(aX + b, cY + d) = Cov(aX + b, cY) + Cov(aX + b, d) = Cov(aX + b, cY) + 0 (Covariance with a constant is 0)

= Cov(aX, cY) + Cov(b, cY) = Cov(aX, cY) + 0

= ac Cov(X, Y)

Cov(aX + b, cY + d) = Cov(aX + b, cY) + Cov(aX + b, d) = ac Cov(X, Y)

and the second statement is not a correct statement.

Cov(X + b, X + d) = Cov(X + b, X) + Cov(X + b, d) = Cov(X + b, X) + 0 (Covariance with a constant is 0)

= Cov(X,X) + Cov(b, X) = Var(X) + 0 = Var(X)

Thus,

Cov(X + b, X + d) = Var(X)    - Correct Statement

Cov(X, c) = 0      - Correct Statement

(c)

The given statement implies independence of Xi, .., Xn. So, the correct statements are

1. Xi, Xj are independent for each i $\ne$ j   

2. For each i, Xi is independent of the collection of r.v. {Xj}

4. EXi Xj = EXi EXj for each i $\ne$ j

5. Cov(Xi, Xj) = 0 for each i $\ne$ j