Question

[1] The joint probability mass function of two discrete random variables A and B is Pab(a,b) = {ca2b, a = -2,2 and b = 1,2 otherwise Clearly stating your reasons, answer the following two (1) Are A and B are uncorrelated? (ii) Are A and B independent? [2] X is continuous uniform (1,7) while Y is exponential with mean 2. If the variance of (X+2Y) is 20, find the correlation coefficient of X and Y.

Answer 1

We would be looking at the first question here as:

Q1) a) The probabilities here are computed in terms of c as:
p(-2, 1) = 4c
p(-2, 2) = 8c
p(2, 1) = 4c
p(2, 2) = 8c

First, we compute the expected values here 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) Here, we have: E(ab) = 0 and also E(a)E(b) = 0.
Therfore a and b are independent variables here.