Mathematical and Applied Statistics Lesson of the Day – The Motivation and Intuition Behind Chebyshev’s Inequality
September 5, 2014 Leave a comment
In 2 recent Statistics Lessons of the Day, I
- introduced Markov’s inequality.
- explained the motivation and intuition behind Markov’s inequality.
Chebyshev’s inequality is just a special version of Markov’s inequality; thus, their motivations and intuitions are similar.
Markov’s inequality roughly says that a random variable is most frequently observed near its expected value, . Remarkably, it quantifies just how often is far away from . Chebyshev’s inequality goes one step further and quantifies that distance between and in terms of the number of standard deviations away from . It roughly says that the probability of being standard deviations away from is at most . Notice that this upper bound decreases as increases – confirming our intuition that it is highly improbable for to be far away from .
As with Markov’s inequality, Chebyshev’s inequality applies to any random variable , as long as and are finite. (Markov’s inequality requires only to be finite.) This is quite a marvelous result!