Popoviciu's inequality on variances
Probability theory upper bound
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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]
Where is the third central moment.
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:
Proof via the Bhatia–Davis inequality
Let be a random variable with mean , variance , and . Then, since ,
.
![{\displaystyle 0\leq \mathbb {E} [(M-A)(A-m)]=-\mathbb {E} [A^{2}]-mM+(m+M)\mu }](http://wikimedia.org/api/rest_v1/media/math/render/svg/64956461713b953eb891b7ed80a4355f790798ff)
Thus,
.
![{\displaystyle \sigma ^{2}=\mathbb {E} [A^{2}]-\mu ^{2}\leq -mM+(m+M)\mu -\mu ^{2}=(M-\mu )(\mu -m)}](http://wikimedia.org/api/rest_v1/media/math/render/svg/cd42cca8e2aef14785e26a6e7443352649d2d66c)
Now, applying the Inequality of arithmetic and geometric means, , with and , yields the desired result:
.
