Prove Jensen's inequality for discrete random variables via induction as follows. For a convex function \(f\) :

(a) Let a random variable \(X\) take on two values \(x_{1}

(b) Assume that \(E[f(X)] \geq f(E[X])\) for any discrete random variable \(X\) that takes on \(N\) values. Show that \(E[f(X)] \geq f(E[X])\) for any discrete random variable \(Y\) that takes on \(N+1\) values.