Popoviciu's inequality on variances

In probability theory, Popoviciu's inequality, named after Tiberiu Popoviciu, is an upper bound on the variance ''σ''² of any bounded probability distribution. Let ''M'' and ''m'' be upper and lower bounds on the values of any random variable with a particular probability distribution. Then Popoviciu's inequality states:

: $\sigma^2 \le \frac14 ( M - m )^2.$

This equality holds precisely when half of the probability is concentrated at each of the two bounds.

Sharma ''et al''. have sharpened Popoviciu's inequality:

: $\le \frac14 (M - m)^2.$

Popoviciu's inequality is weaker than the Bhatia–Davis inequality which states

: $\sigma^2 \le ( M - \mu )( \mu - m )$

where ''μ'' is the expectation of the random variable.

In the case of an independent sample of ''n'' observations from a bounded probability distribution, the von Szokefalvi Nagy inequality gives a lower bound to the variance of the sample mean:

: $\sigma^2 \ge \frac .$

Proof via the Bhatia–Davis inequality

Let $A$ be a random variable with mean $\mu$, variance $\sigma^2$, and $\Pr(m \leq A \leq M) = 1$. Then, since $m \leq A \leq M$,

0 \leq \mathbb M - A)(A - m)= -\mathbb ^2- m M + (m+M)\mu.

Thus,

\sigma^2 = \mathbb ^2- \mu^2 \leq - m M + (m+M)\mu - \mu^2 = (M - \mu) (\mu - m).

Now, applying the Inequality of arithmetic and geometric means, $ab \leq \left( \frac \right)^2$, with $a = M - \mu$ and $b = \mu - m$, yields the desired result:

$\sigma^2 \leq (M - \mu) (\mu - m) \leq \frac$.

References

Theory of probability distributions
Statistical inequalities
Statistical deviation and dispersion

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