Popoviciu's inequality on variances

In probability theory, Popoviciu's inequality, named after Tiberiu Popoviciu[citation needed], is an upper bound on the variance 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]

${\displaystyle {\text{variance}}\leq {\frac {1}{4}}(M-m)^{2}.}$

Sharma et al. have proved an improvement of the Popoviciu's inequality that says that:[2]

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

Equality holds precisely when half of the probability is concentrated at each of the two bounds.

Popoviciu's inequality is weaker than the Bhatia–Davis inequality.

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 bound on variance with applications". Journal of Mathematical Inequalities. 4: 355–363. doi:10.7153/jmi-04-32.