Alec's expected value trick

[ilmath]\newcommand{\P}[2][]{\mathbb{P}#1{\left[{#2}\right]} } \newcommand{\Pcond}[3][]{\mathbb{P}#1{\left[{#2}\!\ \middle\vert\!\ {#3}\right]} } \newcommand{\Plcond}[3][]{\Pcond[#1]{#2}{#3} } \newcommand{\Prcond}[3][]{\Pcond[#1]{#2}{#3} }[/ilmath]

[ilmath]\newcommand{\E}[1]{ {\mathbb{E}{\left[{#1}\right]} } } [/ilmath][ilmath]\newcommand{\Mdm}[1]{\text{Mdm}{\left({#1}\right) } } [/ilmath][ilmath]\newcommand{\Var}[1]{\text{Var}{\left({#1}\right) } } [/ilmath][ilmath]\newcommand{\ncr}[2]{ \vphantom{C}^{#1}\!C_{#2} } [/ilmath]

Lemma

Suppose is a non-negative real random-variable, then I have found a way to calculate the expected value of [ilmath]X[/ilmath], [ilmath]\E{X} [/ilmath]:

• We claim: [math]\E{X}\eq\int^\infty_{0}\big(1-\P{X<x}\big)\mathrm{d}x[/math] or [math]\E{X}\eq\int^\infty_0\P{X\ge x}\mathrm{d}x[/math]

Statement

We should be able to use this for a general real random variable by considering conditional random variables (via conditional probability) for the positive and negative parts respectively.