Cantelli's inequality


In probability theory, Cantelli's inequality is a generalization of Chebyshev's inequality in the case of a single "tail". The inequality states that
where
Combining the cases of and gives, for
While the inequality is often attributed to Francesco Paolo Cantelli who published it in 1928, it originates in Chebyshev's work of 1874. The Chebyshev inequality implies that in any data sample or probability distribution, "nearly all" values are close to the mean in terms of the absolute value of the difference between the points of the data sample and the weighted average of the data sample. The Cantelli inequality gives a way of estimating how the points of the data sample are bigger than or smaller than their weighted average without the two tails of the absolute value estimate. The Chebyshev inequality has "higher moments versions" and "vector versions", and so does the Cantelli inequality.

Proof

Let be a real-valued random variable with finite variance and expectation, and define .
Then, for any, we have
the last inequality being a consequence of Markov's inequality. As the above holds for any choice of, we can choose to apply it with the value that minimizes the function. By differentiating, this can be seen to be, leading to
using the previous derivation on. By taking the complement of the left-hand side, we obtain

Generalizations

Using more moments, various stronger inequalities can be shown.
He, Zhang and Zhang and showed, when
and