Borv

[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] Borvs, (a special case of aMorv) is one of the most fundamental distributions there is and many standard distribution are built from it.

Definition

For [ilmath]X\sim\text{Borv}(p)[/ilmath], for [ilmath]p\in [0,1]\subseteq\mathbb{R} [/ilmath] we have [ilmath]\P{X\eq 1}:\eq p[/ilmath] and [ilmath]\P{X\eq 0}\eq 1-p[/ilmath] - there are no other values [ilmath]X[/ilmath] may take.

For this we have the following properties:

• [ilmath]\E{X}\eq p[/ilmath]
• [ilmath]\E{X^2}\eq p[/ilmath]
• [ilmath]\text{Var}(X)\eq p(1-p)[/ilmath]
• [ilmath]\text{Mdm}(X)\eq 2p(1-p)[/ilmath]
• [ilmath]\text{Mdm}^2(X)\eq 2p\big\vert(1-p)(1-2p)\big\vert[/ilmath]
  • This is [ilmath]0[/ilmath] for [ilmath]p\eq 0[/ilmath], [ilmath]p\eq \frac{1}{2} [/ilmath] and [ilmath]p\eq 1[/ilmath]
  • It is maximised for [ilmath]p\eq \frac{1}{2}\left(1\pm\frac{1}{\sqrt{3} }\right)[/ilmath] or [math]p\eq \frac{1}{2}\pm\frac{\sqrt{3} }{6} [/math]