1. Suppose Jill and Bill are each asked to turn a fortune wheel of known contents once. However, the result of the rolls are not revealed to the contestants. The contestants are then asked to whether take their unknown portion home or equally split what they collect. It is reasonable to assume that the prizes obtained by Jill, X, and Bill, Y, are independent identically distributed random variables drawn from the same set of N distinct prizes of value greater than equal to zero, each with probability pi, i ∈ {1, ···, N}. The mean of the prizes on a roll of the wheel is µ and variance σ2.
a) Suppose the presenter announces the total prize X + Y . Could this information change the probability distribution of the difference between Jill's prize and Bill's prize X - Y ? If yes, describe a fortune wheel where it would. If not, prove that it wouldn't for any fortune 1 wheel with N distinct prizes of value greater than equal to zero, each with probability pi,i ∈ {1, ···,N}.
Suppose there are two random variables A and B. Prove that if knowing B changes the distribution of A conditional on B, (A|B), A and B cannot be independent.)
b) Define the random variable U as U = aX+b, where a> 0and b are deterministic scalars. Compute the correlation coefficient Cor(U,X)= ρU,X = Cov(U,X)/√Var(U) √Var(X)
How does your answer change if a< 0?
c) Compute the correlation coefficient between the total prize and the difference between Jill's prize and Bill's prize in terms of µ and σ2.