Question

Consider the two statements.

S₁ : There exist random variables X and Y such that

(E[X - E(X)) (Y - E(Y))])² > Var[X] Var[Y]

S₂ : For all random variables X and Y,

Cov[X, Y] = E [|X - E[X]| |Y - E[Y]|]

Which one of the following choices is correct?

1. S₁ is false, but S₂ is true.
2. S₁ is true, but S₂ is false.
3.

Both S₁ and S₂ are true.

4. Both S₁ and S₂ are false.

Answer 1

Correct Answer - Option 4 : Both S₁ and S₂ are false.

Answer: Option 4

Formula:

Covariance(cov):

cov(X, Y) = E[ (X - E[X]) ] [ (Y - E[Y]) ] 

Theorem:

If X and Y be Random Variables and Let the variances of X and Y exist and be finite then

cov(X,Y)² ≤ var(X)var(Y)

Statement 1:There exist random variables X and Y such that

(E[X - E(X)) (Y - E(Y))])2 > Var[X] Var[Y]

This is not correct. The Square of covariance is less than equal to the product of variance of the two random variables not greater than.

Statement 2:For all random variables X and Y,

Cov[X, Y] = E [|X - E[X]| |Y - E[Y]|]

This is also not correct because Covariance may result in negative as well but the given expression will generate positive value.