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:
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