Question

Please answer all parts of the question. Thank you

[1] The joint probability mass function of two discrete random variables A and B is a = -2, 2 and b = 1,2 0, otherwise Clearly stating your reasons, answer the following two (i) Are A and B are uncorrelated? (ii) Are A and B independent?

Answer 1

a) First, we compute the value of c here by using the property that the sum of all probabilities in the joint (a, b) space should be 1.

Therefore, we have here:
p(-2, 1) + p(-2, 2) + p(2, 1) + p(2, 2) = 1

4c + 8c + 4c + 8c = 1

24c = 1

c = 1/24

Therefore, the covariance of the two variables here is computed as:
E(ab) = -2p(-2, 1) - 4p(-2, 2) + 2p(2, 1) + 4p(2, 2)
E(ab) = -8c - 32c + 8c + 32c = 0

E(a) = -2*12c + 2*12c = 0
Therefore, Cov(a, b) = E(ab) - E(a)E(b) = 0

As the covariance here is 0, therefore yes A and B are uncorrelated here. (because the numerator for correlation is covariance )

b) From the given obtained probabilities, we have here:
P(a = -2) = 12c = 0.5
P(a = -2 | b = 1) = 4c / 8c = 0.5 which is same as P(a = -2)

Therefore a and b are independent here.