Popoviciu's inequality on variances
In
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
- ^ Popoviciu, T. (1935). "Sur les équations algébriques ayant toutes leurs racines réelles". Mathematica (Cluj). 9: 129–145.
- doi:10.7153/jmi-04-32.)
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: CS1 maint: multiple names: authors list (link - JSTOR 2589180.
- ^ Nagy, Julius (1918). "Über algebraische Gleichungen mit lauter reellen Wurzeln". Jahresbericht der Deutschen Mathematiker-Vereinigung. 27: 37–43.