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