1. State whether the following statements are true or false. If a statement is true, briefly [20 Marks] explain why. If a statement is false, either briefly explain why it is false, or give a counter- example. (Most of your answers should be no more than two lines long.) a) If the correlation between two random variables X and Y is 0, then X and Y must be independent. b) For any events, A and B, such that 0 P(AE )P(E.). e) The function /x(x) = je-, re R is a valid probability density function. ( If X has variance ox , then for fixed n the standard deviation of Y =nX + 1, is DY = nox - g) The cumulant generating function has the identity Kx (0) = 1. h) The function Fx(x) below is a valid cumulative distribution function, Fx(x] = " It is possible to have a random variable with an infinite mean but finite variance. i) It is possible for Fx(x)