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In mathematics, the Bhatia–Davis inequality, named after Rajendra Bhatia and Chandler Davis, is an upper bound on the variance 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:

\( \text{variance} \le (M - \mu)(\mu - m). \, \)

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.
See also

Cramér–Rao bound
Chapman–Robbins bound


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.

Mathematics Encyclopedia

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