If X is a non-negative random variable with E(X) = ?, then Markov's inequality tells us that for every a, P(X ? a) ? ?/a. As we saw in class, this bound is sometimes very loose. In this problem we'll look at a situation in which extra information can be used to tighten the bound.
Suppose that we are told X is bounded from below by some number b, meaning P (X ? b) = 1. Use this information to find a tighter bound on P (X ? a). Hint: Apply Markov's inequality to the r.v. Y = X ? b.
For concreteness, suppose E(X) = 1000 and P(X ? 500) = 1. Markov's says that P(X ? 2000) ? 1000/2000 = 1/2. Use your method find a tighter bound on P (X ? 2000).