Popoviciu's_inequality_on_variances Knowpia

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]

\sigma ^{2}\leq {\frac {1}{4}}(M-m)^{2}.

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]

{\sigma ^{2}+\left({\frac {\text{Third central moment}}{2\sigma ^{2}}}\right)^{2}}\leq {\frac {1}{4}}(M-m)^{2}.

If one additionally assumes knowledge of the expectation, then the stronger Bhatia–Davis inequality holds

\sigma ^{2}\leq (M-\mu )(\mu -m)

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:

\sigma ^{2}\geq {\frac {(M-m)^{2}}{2n}}.

Proof via the Bhatia–Davis inequality edit

Let $A$ be a random variable with mean $\mu$ , variance $\sigma ^{2}$ , and $\Pr(m\leq A\leq M)=1$ . Then, since $m\leq A\leq M$ ,

$0\leq \mathbb {E} [(M-A)(A-m)]=-\mathbb {E} [A^{2}]-mM+(m+M)\mu$ .

Thus,

$\sigma ^{2}=\mathbb {E} [A^{2}]-\mu ^{2}\leq -mM+(m+M)\mu -\mu ^{2}=(M-\mu )(\mu -m)$ .

Now, applying the Inequality of arithmetic and geometric means, $ab\leq \left({\frac {a+b}{2}}\right)^{2}$ , with $a=M-\mu$ and $b=\mu -m$ , yields the desired result:

$\sigma ^{2}\leq (M-\mu )(\mu -m)\leq {\frac {\left(M-m\right)^{2}}{4}}$ .

References edit

^ 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.{{cite journal}}: CS1 maint: multiple names: authors list (link)
^ 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.
^ Nagy, Julius (1918). "Über algebraische Gleichungen mit lauter reellen Wurzeln". Jahresbericht der Deutschen Mathematiker-Vereinigung. 27: 37–43.

[Popoviciu-1] Popoviciu, T. (1935). "Sur les équations algébriques ayant toutes leurs racines réelles". Mathematica (Cluj). 9: 129–145.

[Sharma_et_al.-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.

[Nagy1918-4] Nagy, Julius (1918). "Über algebraische Gleichungen mit lauter reellen Wurzeln". Jahresbericht der Deutschen Mathematiker-Vereinigung. 27: 37–43.

[1]

[2]

[3]

[4]

Summary

Proof via the Bhatia–Davis inequality edit

References edit