Notes: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]

Definition

Let [ilmath]p\in[/ilmath][ilmath][0,1][/ilmath][ilmath]\subseteq\mathbb{R} [/ilmath] be given, this represents the probability of success, then we define a random variable (with [ilmath]p[/ilmath] as a parameter), [ilmath]X[/ilmath], as follows:

• [ilmath]X\sim\text{Borv}(p)[/ilmath]

Such that:

• [ilmath]\P{X\eq 1}:\eq p[/ilmath] - the probability of a "true" or "successful" outcome is [ilmath]p[/ilmath] by definition, and
• [ilmath]\P{X\eq 0}\eq 1-p[/ilmath] - the probability of "false" or a "failed" outcome is [ilmath]1-p[/ilmath]

[ilmath]X[/ilmath] is undefined for everything else

Note that:

• In the definition [ilmath]p[/ilmath] always refers to the "active" outcome like pass, and thus [ilmath]1-p[/ilmath] refers to the "passive" or "do nothing" option of failure.

  This is in contrast to the definition of

Test

• puppies

Notation

Sometimes we use values other than [ilmath]0[/ilmath]