Introduction to Probability · Problem 8.2.5

Central Limit Theorem — Grinstead & Snell

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

Problem

Let

X

be a continuous random variable with values normally distributed over

(-\infty, +\infty)

with mean

\mu = 0

and variance

\sigma^2 = 1

. (a) Using Chebyshev's Inequality, find upper bounds for the following probabilities:

P(|X| \geq 1)

P(|X| \geq 2)

, and

P(|X| \geq 3)

. (b) The area under the normal curve between

-1

and

1

is .6827, between

-2

and

2

is .9545, and between

-3

and

3

it is .9973 (see the table in Appendix A). Compare your bounds in (a) with these exact values. How good is Chebyshev's Inequality in this case?