Chebyshev's inequality

Chebyshev's inequality is a probabilistic inequality. It provides an upper bound to the probability that the absolute deviation of a random variable from its mean will exceed a given threshold.

Table of contents

Statement
Example
Important applications
Solved exercises
1. Exercise 1
2. Exercise 2
Other inequalities

Statement

The following is a formal statement.

Proposition Let be a random variable having finite mean and finite variance . Let (i.e., is a strictly positive real number). Then, the following inequality, called Chebyshev's inequality, holds: [eq2]

The proof is a straightforward application of Markov's inequality.

Proof

Since is a positive random variable, we can apply Markov's inequality to it: [eq4] By setting $c=k^{2}$ , we obtain [eq5] But if and only if , so we can write [eq8] Furthermore, by the very definition of variance, Therefore, [eq10]

Example

Suppose that we extract an individual at random from a population whose members have an average income of $40,000, with a standard deviation of $20,000.

What is the probability of extracting an individual whose income is either less than $10,000 or greater than $70,000?

In the absence of more information about the distribution of income, we cannot compute this probability exactly. However, we can use Chebyshev's inequality to compute an upper bound to it.

If denotes income, then is less than $10,000 or greater than $70,000 if and only ifwhere and .

The probability that this happens is: [eq12]

Therefore, the probability of extracting an individual outside the income range $10,000-$70,000 is less than .

Important applications

Chebyshev's inequality has many applications, but the most important one is probably the proof of a fundamental result in statistics, the so-called Chebyshev's Weak Law of Large Numbers.

Solved exercises

Below you can find some exercises with explained solutions.

Exercise 1

Let be a random variable such that [eq13]

Find a lower bound to its variance.

Solution

The lower bound can be derived thanks to Chebyshev's inequality: [eq14] where: in step we have used the inequality; in step we have used the fact that ; in we have used the formula for the probability of a complement; in step we have used the monotonicity of probability:Thus, the lower bound is

Exercise 2

Let be a random variable such that [eq18]

Find an upper bound to the probability

Solution

We can solve this problem as follows: [eq20] where: in step we have used the monotonicity of probability; in step we have used Chebyshev's inequality.

Other inequalities

If you like this page, StatLect has other pages on probabilistic inequalities:

Chebyshev inequality compared to other inequalitites.

How to cite

Please cite as:

Taboga, Marco (2021). "Chebyshev's inequality", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/fundamentals-of-probability/Chebyshev-inequality.