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]
- [math]\displaystyle{ \sigma^2 \le \frac14 ( M - m )^2. }[/math]
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]
- [math]\displaystyle{ {\sigma^2 + \left( \frac \text {Third central moment} {2\sigma^2} \right)^2} \le \frac14 (M - m)^2. }[/math]
If one additionally assumes knowledge of the expectation, then the stronger Bhatia–Davis inequality holds
- [math]\displaystyle{ \sigma^2 \le ( M - \mu )( \mu - m ) }[/math]
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:
- [math]\displaystyle{ \sigma^2 \ge \frac{ ( M - m )^2} {2n}. }[/math]
Proof via the Bhatia–Davis inequality
Let [math]\displaystyle{ A }[/math] be a random variable with mean [math]\displaystyle{ \mu }[/math], variance [math]\displaystyle{ \sigma^2 }[/math], and [math]\displaystyle{ \Pr(m \leq A \leq M) = 1 }[/math]. Then, since [math]\displaystyle{ m \leq A \leq M }[/math],
[math]\displaystyle{ 0 \leq \mathbb{E}[(M - A)(A - m)] = -\mathbb{E}[A^2] - m M + (m+M)\mu }[/math].
Thus,
[math]\displaystyle{ \sigma^2 = \mathbb{E}[A^2] - \mu^2 \leq - m M + (m+M)\mu - \mu^2 = (M - \mu) (\mu - m) }[/math].
Now, applying the Inequality of arithmetic and geometric means, [math]\displaystyle{ ab \leq \left( \frac{a+b}{2} \right)^2 }[/math], with [math]\displaystyle{ a = M - \mu }[/math] and [math]\displaystyle{ b = \mu - m }[/math], yields the desired result:
[math]\displaystyle{ \sigma^2 \leq (M - \mu) (\mu - m) \leq \frac{\left(M - m\right)^2}{4} }[/math].
References
- ↑ Popoviciu, T. (1935). "Sur les équations algébriques ayant toutes leurs racines réelles". Mathematica (Cluj) 9: 129–145.
- ↑ 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.
- ↑ 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.
- ↑ Nagy, Julius (1918). "Über algebraische Gleichungen mit lauter reellen Wurzeln". Jahresbericht der Deutschen Mathematiker-Vereinigung 27: 37–43. https://eudml.org/doc/145531.
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