Introduction to Probability · Problem 8.1.10

Law of Large Numbers — Grinstead & Snell

Mathematics — Probability · chebyshev-inequality

Problem

Let

X

be any random variable which takes on values

0, 1, 2, \ldots, n

and has

E(X) = V(X) = 1

. Show that, for any positive integer

k

P(X \geq k+1) \leq \frac{1}{k^2}.