# fixedpoint.jp - A proof of the Bhatia-Davis inequality

 Web fixedpoint.jp

## A proof of the Bhatia-Davis inequality

The Bhatia–Davis inequality gives an upper bound on the variance of a bounded random variable. It has the following simple proof.

Let $$X$$ be a bounded random variable with maximum $$M$$ and minimum $$m.$$ Also, let $$\mu$$ be $$X$$'s expected value. Without loss of generality we can assume that $$m = 0.$$ (Otherwise, another random variable $$Y := X-m$$ has the same variance as $$X$$'s, and $$(M-\mu)(\mu-m) = \{(M-m)-(\mu-m)\}\{(\mu-m)-0\}.$$) Then$(M-\mu)(\mu-m)-\mathbb{E}[(X-\mu)^2] = \{(M+m)\mu+\mu^2-Mm\}-(\mathbb{E}[X^2]-\mu^2) = (M+m)\mu-Mm-\mathbb{E}[X^2] = M\mu-\mathbb{E}[X^2] \geq M\mu - M\mu = 0.$QED.