=+17. Let X be a random variable with E(X) = 0 and E(X2) = 2. Show that Pr(X c) a2 + 2 (a

=+17. Let X be a random variable with E(X) = 0 and E(X2) = σ2. Show that Pr(X ≥

c) ≤

a2 + σ2

(a + c)2 (3.9)

for all nonnegative a and

c. Prove that the choice a = σ2/c minimizes the right-hand side of (3.9) and that for this choice Pr(X ≥

c) ≤

σ2

σ2 + c2 .

This is Cantelli’s inequality [60].