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# Bhatia–Davis inequality

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

References

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.

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