Misplaced Pages

Popoviciu's inequality on variances

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.

In probability theory, Popoviciu's inequality, named after Tiberiu Popoviciu, is an upper bound on the variance σ 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:

σ 2 ≤ 1 4 ( M − m ) 2 . {\displaystyle \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:

σ 2 + ( Third central moment 2 σ 2 ) 2 ≤ 1 4 ( M − m ) 2 . {\displaystyle {\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

σ 2 ≤ ( M − μ ) ( μ − m ) {\displaystyle \sigma ^{2}\leq (M-\mu )(\mu -m)}

where μ is the expectation of the random variable.

In the case of an independent sample of n observations from a bounded probability distribution, the von Szokefalvi Nagy inequality gives a lower bound to the variance of the sample mean:

σ 2 ≥ ( M − m ) 2 2 n . {\displaystyle \sigma ^{2}\geq {\frac {(M-m)^{2}}{2n}}.}

Proof via the Bhatia–Davis inequality

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

0 ≤ E [ ( M − A ) ( A − m ) ] = − E [ A 2 ] − m M + ( m + M ) μ {\displaystyle 0\leq \mathbb {E} =-\mathbb {E} -mM+(m+M)\mu } .

Thus,

σ 2 = E [ A 2 ] − μ 2 ≤ − m M + ( m + M ) μ − μ 2 = ( M − μ ) ( μ − m ) {\displaystyle \sigma ^{2}=\mathbb {E} -\mu ^{2}\leq -mM+(m+M)\mu -\mu ^{2}=(M-\mu )(\mu -m)} .

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

σ 2 ≤ ( M − μ ) ( μ − m ) ≤ ( M − m ) 2 4 {\displaystyle \sigma ^{2}\leq (M-\mu )(\mu -m)\leq {\frac {\left(M-m\right)^{2}}{4}}} .

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. 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.


Stub icon

This probability-related article is a stub. You can help Misplaced Pages by expanding it.

Categories: