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:
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:
where n is the sample size.
Popoviciu's inequality is weaker than the Bhatia–Davis inequality which states
where μ is the expectation of the random variable.
A lower bound for the variance based on the Bhatia–Davis inequality has been found by Agarwal et al
- 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 bound on variance with applications". Journal of Mathematical Inequalities. 4: 355–363. doi:10.7153/jmi-04-32.CS1 maint: Multiple names: authors list (link)
- Nagy JVS (1918) Uber algebraische Gleichungen mit lauter reellen Wurzeln, Jahresbericht der deutschen mathematiker-Vereingung, 27:37–43
- Agarwal RP, Barnett NS, Cerone P and Dragomir SS (2005) A survey on some inequalities for expectation and variance. Computers and mathematics with applications 49 (2005) 429-480
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