Bhatia–Davis inequality

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In mathematics, the Bhatia–Davis inequality, named after Rajendra Bhatia and Chandler Davis, is an upper bound on the variance σ² of any bounded probability distribution on the real line.

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.[1]

See also[edit]


  1. ^ 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
  • Bhatia, Rajendra; Davis, Chandler (April 2000). "A Better Bound on the Variance". American Mathematical Monthly. Mathematical Association of America. 107 (4): 353–357. doi:10.2307/2589180. ISSN 0002-9890. JSTOR 2589180.