Jensen's inequality states that for a convex function

f, E[f(x)] = f(E[X]).

(a) Prove Jensen's inequality

(b) If X has mean E[X], show that X E[X], where E[X] is the constant random variable.

(c) Suppose that there exists a random variable Z such that E[ZIY] = O and such that X has the same distribution as Y + Z. Show that X, Y. In fact, it can be shown (though the other direction is difficult to prove) that this is a necessary and sufficient condition for X Z, Y.