Question

10. Suppose that for the next 6 months, each day, you do the following
    
    1) record the stock return of Apple Inc.;
    2) draw a random observation from a Normal Distribution with mean 1 and variance 0;
    3) draw a random observation from a Normal Distribution with mean 0 and variance 1.
    
    Which statement is/are false?
      I) Covariance between 1) and 2) will be exactly zero
    II) Covariance between 1) and 3) will be exactly zero
    III) Covariance between 1) and 3) will be close to zero
    IV) Covariance between 2) and 3) will be exactly to zero
    
    (a) I
    (b) II
    (c) III
    (d) IV
    (e) none of them

Answer 1

If X and Y are two random variable, then covariance of X and Y is given by

If X and Y are two independent random variable, then , therefore

In the given three daily works, all three works recording of stock return of Apple Inc, drawing of random observation from normal distribution with mean 1 and variance 0 and drawing of random observation from normal distribution with mean 0 and variance 1 are independent work or event.

Therefore, Covariance of 1 and 2 will be zero, Covariance of 1 and 3 will be zero and Covariance of 2 and 3 will be zero.

All four statements are correct as covariance is zero in all the cases.

Correct option is (e) None of them(there is no false statement)