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

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.