Notes:Statistical test random variable

$\newcommand{\B}[0]{ {\mathbb{B} } }$$\newcommand{\O}[0]{ {\mathcal{O} } }$

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

$\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} }$

Notice

This actually might just be the definition of joint probability applied to tests....

Starting point

Let $(S,\Omega,\mathbb{P})$ be the probability space we take subjects from (so each subject takes a value in $S$, so forth); then:

• Let: $\mathbb{B}:\eq\{0,\ 1\}$ (or $\mathbb{B}:\eq\{-,\ +\}$) for "negative" and "positive" respectively.
  • We imbue $\B$ with the sigma-algebra $\mathcal{P}(\mathbb{B})$, which is: $\big\{\emptyset,\{0\},\{1\},\{0,1\}\big\}$

We introduce a random variable, $P:S\rightarrow\B$,

• such that: $P:S\rightarrow\B$ is an "oracle" of sorts, specifically we have:
  1. $P:s\mapsto 1$ for a subject with the property being tested for, and,
  2. $P:s\mapsto 0$ if the property is absent.
• We assume $P$ is never wrong.
• Note:
  • The random variable requirements imbue that $P^{-1}(\{i\})\in\Omega$ for $i\in\B$ - this should be enough as per generator of a sigma-algebra (
    TODO: check
    ), but if not implicit we add:
    1. $P^{-1}(\{0,1\})\eq S\in\Omega$ and
    2. $P^{-1}(\emptyset)\eq\emptyset\in\Omega$ too

Step 1

We now introduce another random variable:

• $T:S\rightarrow\B$ - for the same sigma-algebras as already covered, however $T$ need not have any "oracular" properties, it represents our test.

We now introduce:

• $\O:S\rightarrow\B\times\B$ given by $\O:s\mapsto\big(P(s),T(s)\big)$
  • We claim this is a random variable itself.
  • We claim that: $\O^{-1}(\{i,j\})\eq P^{-1}(\{i\})\cap T^{-1}(\{j\})$

Finally

We can now talk about $\P{P\eq i\text{ and }T\eq j}$, thus about conditional probabilities like $\Pcond{P\eq 1}{T\eq 1}$ as a result - which is the goal.