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

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(wikilink for Tiberiu Popoviciu)
 
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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}}
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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}}
  
 
: <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.

Template:Probability-stub