# Mathematical and Applied Statistics Lesson of the Day – The Motivation and Intuition Behind Markov’s Inequality

August 27, 2014 4 Comments

Markov’s inequality may seem like a rather arbitrary pair of mathematical expressions that are coincidentally related to each other by an inequality sign:

where .

However, there is a practical motivation behind Markov’s inequality, and it can be posed in the form of a simple question: How often is the random variable “far” away from its “centre” or “central value”?

Intuitively, the “central value” of is the value that of that is most commonly (or most *frequently*) observed. Thus, as deviates farther and farther from its “central value”, we would expect those distant-from-the-centre values to be less frequently observed.

Recall that the **expected value**, , is a measure of the “centre” of . Thus, we would expect that the probability of being very far away from is very low. Indeed, Markov’s inequality rigorously confirms this intuition; here is its rough translation:

As becomes really far away from , the event becomes less probable.

You can confirm this by substituting several key values of .

- If , then ; this is the highest upper bound that can get. This makes intuitive sense; is going to be frequently observed near its own expected value.

- If , then . By Kolmogorov’s axioms of probability, any probability must be inclusively between and , so . This makes intuitive sense; there is no possible way that can be bigger than positive infinity.

$c < E[X]$ gives an upper bound greater than 1, but obviously the upper bound is 1 since its a probability function

Hello J,

I don’t fully understand your point in the first part of your comment. Could you please elaborate?

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