(One-sided Chebyshev bound) Let X be a random variable such that E(X) = 0 and Var(X) = ????2. Show that, for any a > 0,

This result is known as Cantelli’s inequality.
(Hint: For b > 0, put W = (X + b)2 and apply Markov’s inequality to W; then examine which value of b minimizes the quantity on the right.)

P(X a) 02