Bhatia–Davis inequality
Introduction In mathematics, the '''Bhatia–Davis inequality''', named after Rajendra Bhatia and Chandler Davis, is an upper bound on the variance ''σ''2 of any bounded probability distribution on the real line.
Statement Let ''m'' and M be the lower and upper bounds, respectively, for a set of real numbers ''a1'', ..., ''an ,'' with a particular probability distribution. Let ''μ'' be the expected value of this distribution.Then the Bhatia–Davis inequality states:: Equality holds if and only if every ''aj'' in the set of values is equal either to ''M'' or to ''m''.
Proof Since , .Thus, .
Extensions of the Bhatia–Davis inequality If  is a positive and unital linear mapping of a ''C* -''algebra  into a ''C*'' -algebra , and ''A'' is a self-adjoint element of  satisfying ''m''  ''A''  ''M'', then:.If  is a discrete random variable such that where , then:,where  and .
Comparisons to other inequalities The Bhatia–Davis inequality is stronger than Popoviciu's inequality on variances (note, however, that Popoviciu's inequality does not require knowledge of the expectation or mean), as can be seen from the conditions for equality. Equality holds in Popoviciu's inequality if and only if half of the ''aj''  are equal to the upper bounds and half of the ''aj''  are equal to the lower bounds. Additionally, Sharma has made further refinements on the Bhatia–Davis inequality.
See also *Cramér–Rao bound*Chapman–Robbins bound*Popoviciu's inequality on variances
References