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(view preamble)
Cross-references: line, convex, mean, expected value, range, segment, convex function, random variable, equivalent
This is version 3 of proof of Jensen's inequality, born on 2002-06-06, modified 2006-11-05.
Object id is 3060, canonical name is ProofOfJensensInequality.
Accessed 15882 times total.
Classification:
| AMS MSC: | 39B62 (Difference and functional equations :: Functional equations and inequalities :: Functional inequalities, including subadditivity, convexity, etc.) | | | 26D15 (Real functions :: Inequalities :: Inequalities for sums, series and integrals) |
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Pending Errata and Addenda
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![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
![$\displaystyle \operatorname{\mathbb{E}}[f(X)] \ge f(\operatorname{\mathbb{E}}[X]). $](http://images.planetmath.org:8080/cache/objects/3060/l2h/img7.png "$\displaystyle \operatorname{\mathbb{E}}[f(X)] \ge f(\operatorname{\mathbb{E}}[X]). $"){kind=small}
![$ c=\operatorname{\mathbb{E}}[X]$](http://images.planetmath.org:8080/cache/objects/3060/l2h/img8.png "$ c=\operatorname{\mathbb{E}}[X]$"){kind=small}
![$\displaystyle \operatorname{\mathbb{E}}[f(X)] \ge \operatorname{\mathbb{E}}[\varphi(X)] = \operatorname{\mathbb{E}}[\alpha (X-c) + f(c)] = f(c) $](http://images.planetmath.org:8080/cache/objects/3060/l2h/img15.png "$\displaystyle \operatorname{\mathbb{E}}[f(X)] \ge \operatorname{\mathbb{E}}[\varphi(X)] = \operatorname{\mathbb{E}}[\alpha (X-c) + f(c)] = f(c) $"){kind=small}