In probability theory, Popoviciu's inequality, named after Tiberiu Popoviciu, is an upper bound on the variance σ2 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:[1]
\sigma2\le
| 14 | |
| ( |
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:[2]
{\sigma2+\left(
| Thirdcentralmoment | |
| 2\sigma2 |
\right)2}\le
| 14 | |
| (M |
-m)2.
If one additionally assumes knowledge of the expectation, then the stronger Bhatia - Davis inequality holds
\sigma2\le(M-\mu)(\mu-m)
where μ is the expectation of the random variable.[3]
In the case of an independent sample of n observations from a bounded probability distribution, the von Szokefalvi Nagy inequality[4] gives a lower bound to the variance of the sample mean:
\sigma2\ge
| (M-m)2 | |
| 2n |
.
Let
A
\mu
\sigma2
\Pr(m\leqA\leqM)=1
m\leqA\leqM
0\leqE[(M-A)(A-m)]=-E[A2]-mM+(m+M)\mu
Thus,
\sigma2=E[A2]-\mu2\leq-mM+(m+M)\mu-\mu2=(M-\mu)(\mu-m)
Now, applying the Inequality of arithmetic and geometric means,
ab\leq\left(
| a+b | |
| 2 |
\right)2
a=M-\mu
b=\mu-m
\sigma2\leq(M-\mu)(\mu-m)\leq
| \left(M-m\right)2 | |
| 4 |