 StatLect

# 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

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: 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 that where is the indicator of the event and is the indicator of the event . As a consequence, we can write 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 that Furthermore, 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 obtain The above inequalities can be put together: 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 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: Therefore, the probability of extracting an individual having an income greater than \$200,000 is less than .

## 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: Multiplying both sides of the inequality by , we obtain Adding to both sides of the inequality, we obtain Thus, the lower bound is The book

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