In probability theory, Popoviciu's inequality, named after Tiberiu Popoviciu, is an upper bound on the variance σ2 of any bounded probability distribution.
Let M and m be upper and lower bounds on the values of any random variable with a particular probability distribution.
Then Popoviciu's inequality states:[1] This equality holds precisely when half of the probability is concentrated at each of the two bounds.
Sharma et al. have sharpened Popoviciu's inequality:[2] If one additionally assumes knowledge of the expectation, then the stronger Bhatia–Davis inequality holds where μ is the expectation of the random variable.
[3] In the case of an independent sample of n observations from a bounded probability distribution, the von Szokefalvi Nagy inequality[4] gives a lower bound to the variance of the sample mean: Let
be a random variable with mean
μ
) μ
{\displaystyle 0\leq \mathbb {E} [(M-A)(A-m)]=-\mathbb {E} [A^{2}]-mM+(m+M)\mu }
μ
) μ −
μ
Now, applying the Inequality of arithmetic and geometric means,
, yields the desired result:
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