Chebyshev inequality

Markov inequality placed an upper bound on the probability for a nonnegative random variable X to be larger than a positive number a. Chebyshev inequality does something similar, but with the constraint of nonnegative-ness removed. The price to be paid is that the r.v. under consideration is now required to have a variance (which Markov inequality did not require).

Chebyshev inequality exists in two equivalent and nearly identical forms.

First form

For any r.v. X with mean µ and variance σ², and for any positive number k :

Second form

For any r.v. X with mean µ and variance σ², and for any positive number k :
 

This second form uses the standard deviation as a "measuring stick" to express the distance between a realization of X and the mean of X. It reads : "The probability for a realization of X to fall more than k times the standard deviation away from the mean is bounded by 1/k².".

We demonstrate here Chebyshev inequality.

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Chebyshev inequality will be the key to the Weak Law of Large Numbers.

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