Markov's inequality

Markov's inequality is a probabilistic inequality. It provides an upper bound to the probability that the realization of a random variable exceeds a given threshold.

Table of contents

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

Statement

The proposition below formally states the inequality.

Proposition Let be an integrable random variable defined on a sample space . Let for all (i.e., is a positive random variable). Let (i.e., is a strictly positive real number). Then, the following inequality, called Markov's inequality, holds: [eq3]

Reading and understanding the proof of Markov's inequality is highly recommended because it is an interesting application of many elementary properties of the expected value.

Proof

First note thatwhere is the indicator of the event and is the indicator of the event . As a consequence, we can write [eq9] Now, note that is a positive random variable and that the expected value of a positive random variable is positive:Therefore,Now, note that the random variable is smaller than the random variable for any :because, trivially, is always smaller than when the indicator is not zero. Thus, by an elementary property of the expected value, we have thatFurthermore, by using the linearity of the expected value and the fact that the expected value of an indicator is equal to the probability of the event it indicates, we obtainThe above inequalities can be put together: [eq19] Finally, since is strictly positive we can divide both sides of the right-hand inequality to obtain Markov's inequality:

This property also holds when almost surely (in other words, there exists a zero-probability event such that ).

Example

Suppose that an individual is extracted at random from a population of individuals having an average yearly income of $40,000.

What is the probability that the extracted individual's income is greater than $200,000?

In the absence of more information about the distribution of income, we can use Markov's inequality to calculate an upper bound to this probability: [eq22] Therefore, the probability of extracting an individual having an income greater than $200,000 is less than .

Important applications

Markov's inequality has several applications in probability and statistics.

For example, it is used:

to prove Chebyshev's inequality;
in the proof that mean square convergence implies convergence in probability;
to derive upper bounds on tail probabilities (Exercise 2 below).

Solved exercises

Below you can find some exercises with explained solutions.

Exercise 1

Let be a positive random variable whose expected value is

Find a lower bound to the probability

Solution

First of all, we need to use the formula for the probability of a complement:Now, we can use Markov's inequality: [eq26] Multiplying both sides of the inequality by , we obtainAdding to both sides of the inequality, we obtain

Thus, the lower bound is

Exercise 2

Let be a random variable such that the expected value exists and is finite.

Use the latter expected value to derive an upper bound to the tail probabilitywhere is a positive constant.

Solution

By Markov's inequality, we have [eq32]

Other inequalities

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

Markov inequality compared to other inequalitites.

How to cite

Please cite as:

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