Bhatia–Davis inequality

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:

$\sigma ^{2}\leq (M-\mu )(\mu -m).\,$

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. 107 (4). Mathematical Association of America: 353–357. doi:10.2307/2589180. ISSN 0002-9890. JSTOR 2589180.