Introduction to Probability · Problem 8.2.7

Central Limit Theorem — Grinstead & Snell

Mathematics — Probability · law-of-large-numbers-continuous

Problem

X

is a random variable with mean

\mu \neq 0

and variance

\sigma^2

, define the relative deviation

D

X

from its mean by

D = \left|\frac{X - \mu}{\mu}\right| .

(a) Show that

P(D \geq a) \leq \sigma^2/(\mu^2 a^2)

. (b) If

X

is the random variable of Exercise 1, find an upper bound for

P(D \geq .2)

P(D \geq .5)

P(D \geq .9)

, and

P(D \geq 2)