Mdm of the Binomial distribution

[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]

Statement

Let [ilmath]n\in\mathbb{N}_{\ge 1} [/ilmath] and [ilmath]p\in[0,1]\subseteq[/ilmath][ilmath]\mathbb{R} [/ilmath] and:

• [ilmath]X\sim[/ilmath][ilmath]\text{Bin} [/ilmath][ilmath](n,p)[/ilmath] - so [ilmath]X[/ilmath] is a Binomial random variable

We will calculate the Mdm of [ilmath]X[/ilmath], [ilmath]\E{\big\vert X-\E{X}\big\vert} [/ilmath]

Proof

• [ilmath]\Mdm(X)[/ilmath]