Definitions for cantelli's inequality
can·tel·li's in·equal·i·ty

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1. 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 Pr ( X − E [ X ] ≥ λ ) { ≤ σ 2 σ 2 + λ 2 if λ > 0 , ≥ 1 − σ 2 σ 2 + λ 2 if λ < 0. {\displaystyle \Pr(X-\mathbb {E} [X]\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 X {\displaystyle X} is a real-valued random variable, Pr {\displaystyle \Pr } is the probability measure, E [ X ] {\displaystyle \mathbb {E} [X]} is the expected value of X {\displaystyle X} , σ 2 {\displaystyle \sigma ^{2}} is the variance of X {\displaystyle X} .Combining the cases of λ > 0 {\displaystyle \lambda >0} and λ < 0 {\displaystyle \lambda <0} gives, for δ > 0 , {\displaystyle \delta >0,} Pr ( | X − E [ X ] | ≥ δ ) ≤ 2 σ 2 σ 2 + δ 2 . {\displaystyle \Pr(|X-\mathbb {E} [X]|\geq \delta )\leq {\frac {2\sigma ^{2}}{\sigma ^{2}+\delta ^{2}}}.} 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 (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.

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Numerology

1. Chaldean Numerology

   The numerical value of cantelli's inequality in Chaldean Numerology is: 2

2. Pythagorean Numerology

   The numerical value of cantelli's inequality in Pythagorean Numerology is: 3

References

1. ^ Wikipedia
   https://en.wikipedia.org/wiki/Cantelli'S_Inequality