Notes:Borv

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$\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} }$

Definition

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

• $X\sim\text{Borv}(p)$

Such that:

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

$X$ is undefined for everything else

Note that:

• In the definition $p$ always refers to the "active" outcome like pass, and thus $1-p$ 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 $0$