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Suppose a distribution has minimum m, maximum M, and expected value μ. Then the inequality says:
Equality holds precisely if all of the probability is concentrated at the endpoints m and M.
The Bhatia–Davis inequality is stronger than Popoviciu's inequality on variances.
A lower bound for the variance based on the Bhatia–Davis inequality has been found by Agarwal et al.
- 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