answer all questions with detailed explanation

1. Check whether each of the following claims is true or false. If the claim is true, prove it; otherwise, give a counter example to show that it is indeed false. (a) When two random variables X and Y are independent of each other, they are uncorrelated as well. (5 points) (b) Under the assumption of the linear model that E(6|x) = 0, E is independent of x. (5 points) (c) Under the same assumption as in (b), e and x are uncorrelated. (5 points)