Cantelli's inequality
In probability theory, Cantelli's inequality (also called the Chebyshev-Cantelli inequality and the one-sided Chebyshev inequality) is an improved version of Chebyshev's inequality for one-sided tail bounds. The inequality states that, for λ > 0 , {\displaystyle \lambda >0,} Pr ( X − E [ X ] ≥ λ ) ≤ σ 2 σ 2 + λ 2 , {\displaystyle \Pr(X-\mathbb {E} [X]\geq \lambda )\leq {\frac {\sigma ^{2}}{\sigma ^{2}+\lambda ^{2}}},} 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} .