Popoviciu's inequality on variances

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  , variance  , and  . Then, since  ,

 .

Thus,

 .

Now, applying the Inequality of arithmetic and geometric means,  , with   and  , yields the desired result:

 .

References

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  1. ^ Popoviciu, T. (1935). "Sur les équations algébriques ayant toutes leurs racines réelles". Mathematica (Cluj). 9: 129–145.
  2. ^ Sharma, R., Gupta, M., Kapoor, G. (2010). "Some better bounds on the variance with applications". Journal of Mathematical Inequalities. 4 (3): 355–363. doi:10.7153/jmi-04-32.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  3. ^ Bhatia, Rajendra; Davis, Chandler (April 2000). "A Better Bound on the Variance". American Mathematical Monthly. 107 (4). Mathematical Association of America: 353–357. doi:10.2307/2589180. ISSN 0002-9890. JSTOR 2589180.
  4. ^ Nagy, Julius (1918). "Über algebraische Gleichungen mit lauter reellen Wurzeln". Jahresbericht der Deutschen Mathematiker-Vereinigung. 27: 37–43.