In probability theory, Cantelli's inequality, named after Francesco Paolo Cantelli, is a generalization of Chebyshev's inequality in the case of a single "tail". The inequality states that
where
Combining the cases of
The inequality is due to Francesco Paolo Cantelli. 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 (sometimes called the "Chebyshev–Cantelli inequality" or the "one-sided Chebyshev 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
Then, for any
the last inequality being a consequence of Markov's inequality. As the above holds for any choice of
using the previous derivation on