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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.

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Statement

The proposition below formally states the inequality.

Proposition Let X be an integrable random variable defined on a sample space Omega. Let [eq1] for all omega in Omega (i.e., X is a positive random variable). Let [eq2] (i.e., $c$ 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 that[eq4]where [eq5] is the indicator of the event [eq6] and [eq7] is the indicator of the event [eq8]. As a consequence, we can write[eq9]Now, note that [eq10] is a positive random variable and that the expected value of a positive random variable is positive:[eq11]Therefore,[eq12]Now, note that the random variable [eq13] is smaller than the random variable [eq14] for any omega in Omega:[eq15]because, trivially, $c$ is always smaller than X when the indicator [eq16] is not zero. Thus, by an elementary property of the expected value, we have that[eq17]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[eq18]The above inequalities can be put together:[eq19]Finally, since $c$ is strictly positive we can divide both sides of the right-hand inequality to obtain Markov's inequality:[eq20]

This property also holds when $Xgeq 0$ almost surely (in other words, there exists a zero-probability event E such that [eq21]).

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:[eq22]Therefore, the probability of extracting an individual having an income greater than $200,000 is less than $1/5$.

Solved exercises

Below you can find some exercises with explained solutions.

Exercise 1

Let X be a positive random variable whose expected value is[eq23]Find a lower bound to the probability[eq24]

Solution

First of all, we need to use the formula for the probability of a complement:[eq25]Now, we can use Markov's inequality:[eq26]Multiplying both sides of the inequality by $-1$, we obtain[eq27]Adding 1 to both sides of the inequality, we obtain

[eq28]Thus, the lower bound is[eq29]

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