# Difference between revisions of "Popoviciu's inequality on variances"

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− | In [[probability theory]], '''Popoviciu's inequality''', named after Tiberiu Popoviciu{{Citation needed|date=August 2009}}, is an [[upper bound]] on the [[variance]] 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:{{Citation needed|date=August 2009}} | + | In [[probability theory]], '''Popoviciu's inequality''', named after [[Tiberiu Popoviciu]]{{Citation needed|date=August 2009}}, is an [[upper bound]] on the [[variance]] 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:{{Citation needed|date=August 2009}} |

: <math> \text{variance} \le \frac14 (M - m)^2. </math> | : <math> \text{variance} \le \frac14 (M - m)^2. </math> |

## Latest revision as of 01:01, 4 July 2013

In probability theory, **Popoviciu's inequality**, named after Tiberiu Popoviciu{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B=
{{#invoke:Category handler|main}}{{#invoke:Category handler|main}}^{[citation needed]}
}}, is an upper bound on the variance of any bounded probability distribution. Let *M* and *m* be upper and lower bounds on the values of any random variable with a particular probability distribution. Then Popoviciu's inequality states:{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B=
{{#invoke:Category handler|main}}{{#invoke:Category handler|main}}^{[citation needed]}
}}

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

Popoviciu's inequality is weaker than the Bhatia–Davis inequality.