A random variable X has the following probability function

(i) Find the expected value of X

(a) from Definition 4.4,

(b) using Proposition 4.13, and verify that the two results agree.
(ii) Find the percentage error in the upper bound obtained from Markov’s inequality as an estimate for the following probabilities associated with X:
P(X ≥ 1), P(X ≥ 2), P(X ≥ 3).

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3/4, x = 0, f(x)=1/6, x = 2, 1/12, x = 4.