=+18. Suppose g(x) is a function such that g(x) ≤ 1 for all x and g(x) ≤ 0 for x ≤

c. Demonstrate the inequality Pr(X ≥

c) ≥ E[g(X)] (3.10)

72 3. Convexity, Optimization, and Inequalities for any random variable X [60]. Verify that the polynomial g(x) = (x − c)(c + 2d − x)

d2 with d > 0 satisfies the stated conditions leading to inequality (3.10).

If X is nonnegative with E(X) = 1 and E(X2) = β and c ∈ (0, 1), then prove that the choice d = β/(1 −

c) yields Pr(X ≥

c) ≥ (1 − c)2

β .

Finally, if E(X2) = 1 and E(X4) = β, show that Pr(|X| ≥

c) ≥ (1 − c2)2

β .