Prove a If X and Y are independent random variables the Corr

Prove:
a) If X and Y are independent random variables the Corr(X,Y)=0.

Solution

a)

The table on the left shows the joint probability distribution between two random variables - X and Y; and the table on the right shows the joint probability distribution between two random variables - A and B.

Which of the following statements are true?

I. X and Y are independent random variables.
II. A and B are independent random variables.

(A) I only
(B) II only
(C) I and II
(D) Neither statement is true.
(E) It is not possible to answer this question, based on the information given.

Solution

The correct answer is A. The solution requires several computations to test the independence of random variables. Those computations are shown below.

X and Y are independent if P(x|y) = P(x), for all values of X and Y. From the probability distribution table, we know the following:

P(x=0) = 0.2;      P(x=0 | y=3) = 0.2;      P(x=0 | y = 4) = 0.2
P(x=1) = 0.4;      P(x=1 | y=3) = 0.4;      P(x=1 | y = 4) = 0.4
P(x=2) = 0.4;      P(x=2 | y=3) = 0.4;      P(x=2 | y = 4) = 0.4

Thus, P(x|y) = P(x), for all values of X and Y, which means that X and Y are independent. We repeat the same analysis to test the independence of A and B.

P(a=0) = 0.3;      P(a=0 | b=3) = 0.2;      P(a=0 | b = 4) = 0.4
P(a=1) = 0.4;      P(a=1 | b=3) = 0.4;      P(a=1 | b = 4) = 0.4
P(a=2) = 0.3;      P(a=2 | b=3) = 0.4;      P(a=2 | b = 4) = 0.2

Thus, P(a|b) is not equal to P(a), for all values of A and B. For example, P(a=0) = 0.3; but P(a=0 | b=3) = 0.2. This means that A and B are not independent.