(a) Let X be a discrete random variable taking non-negative values; i.e. Rx = No = {0, 1, 2, ...}. Let a > 0. Show that E[X] > aP(X > a) and deduce that P(X > a) < E[X]/a, an inequality known as Markov's inequality. (b) Let X be a discrete random variable. By applying Markov's inequality to a well-chosen random variable, prove Chebyshev's inequality; that is, show that if a > 0 then Var[X] P (|X – EX|> a)< (c) Suppose that F ~ Bin(2m + 1, 5). Use Chebyshev's inequality to bound P[F > m]. From (c), deduce that the probability of a decoding error in the binary sym- metric channel (BSC) with symbol error probability p = can be made arbitrarily small by using a sufficiently long repetition code. сan

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