Popoviciu's inequality on variances

In probability theory, Popoviciu's inequality, named after Tiberiu Popoviciu, 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]

{\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]

{{\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

↑ 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.

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