1 Just One Tail, Please Let X be some random variable with nite mean and variance which is not necessarily non- negative. The extended version of Markov's Inequality states that for a non-negative, monoton- ically increasing function ([5 (X) and 06 > O, P(X>a)gw _ (PW) Suppose E[X] = O, Var(X) = 0'2 0. (a) Use the extended version of Markov's Inequality stated above with q) (x) = (x l (3)2, where c is some constant, to show that: 2 2P' (X Z a) S 0' + c ( 06 + c)2 (b) Note that the above bound applies for all positive c, so we can choose a value of c to minimize the expression, yielding the best possible bound. Find the value for c which will minimize the RHS expression (you may assume that the expression has a unique minimum). Plug in the minimizing value of c to prove the following bound: 2 o PX>oc