This question asks you to show that the Markov and Chebyshev inequalities are sharp: that is, that there exist cases in which these inequalities hold with equality.

a) Consider a random variable X with a Bernoulli distribution: X = 0 with probability 1-p, and X = 1 with probability p. Let g(X) be a non-negative function of X (and note that, given our X, only the values of g(0) and g(1) matter). Show the following claim: for every constant c > 0, there exists a function g(x) such that the Markov bound on P(g(X) c) holds with equality.

b) Consider a random variable X with the following distribution: X = -1 with probability 0.5a², X = 0 with probability 1-a², and X = 1 with probability 0.5a², where a < 1 is a parameter. Show the following claim: for every constant k > 1, there exists a parameter a > 0 such that the Chebyshev bound on

P(|(X-)/| k) holds with equality.