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:

Proof via the Bhatia–Davis inequality

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

  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. Mathematical Association of America. 107 (4): 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.


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