Expected value: Difference between revisions

Revision as of 00:46, 14 March 2013

Revision as of 17:31, 21 March 2013

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This is sometimes called the [[law of the unconscious statistician]]. Using representations as [[Riemann–Stieltjes integral]] and [[integration by parts]] the formula can be restated as

: <math>\operatorname{E}[g(X)] = \int_aint_{-\infty}^\infty g(x) \, \mathrm{d} \operatorname{P}(X \le x)= \begin{cases} g(a)+ \int_a^\infty g'(x)\operatorname{P}(X > x) \, \mathrm{d} x & \mathrm{if}\ \operatorname{P}(g(X) \ge g(a))=1 \\ g(b) - \int_{-\infty}^b g'(x)\operatorname{P}(X \le x) \, \mathrm{d} x & \mathrm{if}\ \operatorname{P}(g(X) \le g(b))=1. \end{cases}</math>

As a special case let ''α'' denote a positive real number. Then