Question

1) Suppose that X and Y are two random variables, which may be dependent and Var(X)=Var(Y). Assume that 0<Var(X+Y)<∞ and 0<Var(X-Y)<∞. Which of the following statements are NOT true? (There may be more than one correct answer)

a. E(XY) = E(X)E(Y)

b. E(X/Y) = E(X)/E(Y)

c. (X+Y) and (X-Y) are correlated

d. (X+Y) and (X-Y) are not correlated.

2) S.D(X ± Y) is equal to, where S.D means standard deviation

a. S.D(X) ± S.D(Y)

b. Var(X) ± Var(Y)

c. Square root of Var(X) ± Var(Y)

d. Unable to tell

Answer 1

Solution:

Consider the following two samples:

Y	X	X+Y	X - Y
1	25	25 + 1 = 26	25 -1 = 24
2	26	26 + 2 = 28	26 - 2 = 24
3	27	27 + 3 = 30	27 - 3 = 24

Both the samples have variance of 1, so V(X) = V(Y). Further, E(Y) = 2 and E(X) = 26

Now, evaluating further we can answer the given questions.

1) E(XY) = (1*25 + 2*26 + 3*27)/3 = 52.67

E(X)*E(Y) = 26*2 = 52

So, the two aren't exactly same. Also, Cov(X, Y) = E(XY) - E(X)E(Y)

So, if two equal, Cov(X, Y) = 0, which is not necessarily true as X and Y might be dependent

Similarly, E(X/Y) = (25/1 + 26/2 + 27/3)/3 = 47/3 = 15.67

However, E(X)/E(Y) = 26/2 = 13, again the two aren't equal

Finally, (X+Y) changes by 2 but (X - Y) is still constant, explaining that change in (X +Y) does not change (X - Y), and so the two might not be correlated.

Thus, 3 statements are not true: (a), (b) and (c)

2) S.D(X +/- Y) = (V(X +/- Y))^1/2

V(X +/- Y) = V(X) + V(Y) - 2*Cov(X, Y)

So, S.D(X +/- Y) = (V(X) + V(Y) - 2*Cov(X, Y))^1/2

Thus, no option define it correctly

Correct option is (d) unable to tell