Borv

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

$\newcommand{\E}[1]{ {\mathbb{E}{\left[{#1}\right]} } }$$\newcommand{\Mdm}[1]{\text{Mdm}{\left({#1}\right) } }$$\newcommand{\Var}[1]{\text{Var}{\left({#1}\right) } }$$\newcommand{\ncr}[2]{ \vphantom{C}^{#1}\!C_{#2} }$ 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 $X\sim\text{Borv}(p)$, for $p\in [0,1]\subseteq\mathbb{R}$ we have $\P{X\eq 1}:\eq p$ and $\P{X\eq 0}\eq 1-p$ - there are no other values $X$ may take.

For this we have the following properties:

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