Mdm of the normal distribution

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Just starting to ask the question. Once this is worked out, it'll then be changed to a statement/proof page and marked as a theorem Alec (talk) 23:01, 24 November 2017 (UTC)
[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]



Let [ilmath]X\sim[/ilmath][ilmath]\text{Nor} [/ilmath][ilmath](\mu,\sigma^2)[/ilmath], then:

  • [math]\mathbb{E}\big[\ \vert X-\mathbb{E}[X]\vert\ \big]\eq \int_{-\infty}^{+\infty}\big( \vert x-\mathbb{E}[X]\vert\!\ f_X(x)\big)\mathrm{d}x[/math]
    • [math]\eq 2 \int_\mu^\infty\big( (x-\mathbb{E}[X])\!\ f_X(x)\big)\mathrm{d}x[/math] - check this, hopefully reduces from here!