# Cantelli's inequality

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".[1][2][3] The inequality states that

${\displaystyle \Pr(X-\mu \geq \lambda )\quad {\begin{cases}\leq {\frac {\sigma ^{2}}{\sigma ^{2}+\lambda ^{2}}}&{\text{if }}\lambda >0,\\[8pt]\geq 1-{\frac {\sigma ^{2}}{\sigma ^{2}+\lambda ^{2}}}&{\text{if }}\lambda <0.\end{cases}}}$

where

${\displaystyle X}$ is a real-valued random variable,
${\displaystyle \Pr }$ is the probability measure,
${\displaystyle \mu }$ is the expected value of ${\displaystyle X}$,
${\displaystyle \sigma ^{2}}$ is the variance of ${\displaystyle X}$.

With the same methods one can prove the following variant:

${\displaystyle \Pr(|X-\mu |\geq \lambda )\leq {\frac {2\sigma ^{2}}{\sigma ^{2}+\lambda ^{2}}}.}$

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.

## References

1. Research and practice in multiple criteria decision making: proceedings of the XIVth International Conference on Multiple Criteria Decision Making (MCDM), Charlottesville, Virginia, USA, June 8–12, 1998, edited by Y.Y. Haimes and R.E. Steuer, Springer, 2000, ISBN 3540672664.
2. "Tail and Concentration Inequalities" by Hung Q. Ngo
3. "Concentration-of-measure inequalities" by Gábor Lugois