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.
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.
[1]
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
[2] has made further refinements on the Bhatia–Davis inequality.
See also
References