Question 1: Random Variables [25 marks] Two discrete random variables X and Y have support sets \\' = {0, 1} and ) = {0, 1}, respectively. The values of the joint probability mass function (pmf) Px.(x, y) describing X and Y are given by the following table: X 0 O (1 - 6)2 26 (1 - 5) 0 where it is assumed that o e R, 6 0 and o # 1. a) Discuss what is the valid range of 6. [5 marks] b) Marginalise the joint pmf to obtain the pmf's of X and Y . [5 marks] c) Discuss whether X and Y are independent random variables or not. [5 marks] d) Can the joint entropy H(X, Y) equal the sum of the entropies of X and Y (i.e., H(X) and H(Y)? Explain why, or why not, solely in terms of: your result in c), the chain rule of entropy and the conditional entropy inequality. [5 marks] e) Find the value of o that maximises the entropy of X, H(X ), and then find the value of o that maximises the entropy of Y, H(Y). [5 marks]