chore: clarify random variable

src/posts/2025-02-16-probability-distributions.md

@@ -22,13 +22,16 @@ allow Pandoc to render them to the web.
 First, some brief exposition on random variables. Quixotically, a random
 variable is actually a function.
 
-Standard notation, $\Omega$ is sample space, $\omega$ is an event.
+Standard notation: $\Omega$ is a sample space, $\omega \in \Omega$ is an
+event.
 
 *Definition. *
 
 A **random variable** $X$ is a function
-$X:\Omega \rightarrow {\mathbb{R}}$ that gives the probability of an
-event $\omega \in \Omega$.
+$X:\Omega \rightarrow {\mathbb{R}}$ that takes the set of possible
+outcomes in a sample space, and maps it to a [measurable
+space](https://en.wikipedia.org/wiki/Measurable_space), typically (as in
+our case) a subset of $\mathbb{R}$.
 
 *Definition. *
 

typst/2025-02-16-probability-distributions.typ

@@ -29,11 +29,13 @@ render them.
 First, some brief exposition on random variables. Quixotically, a random
 variable is actually a function.
 
-Standard notation, $Omega$ is sample space, $omega$ is an event.
+Standard notation: $Omega$ is a sample space, $omega in Omega$ is an event.
 
 #definition[
-  A *random variable* $X$ is a function $X : Omega -> RR$ that gives the
-  probability of an event $omega in Omega$.
+  A *random variable* $X$ is a function $X : Omega -> RR$ that takes the set of
+  possible outcomes in a sample space, and maps it to a
+  #link("https://en.wikipedia.org/wiki/Measurable_space")[measurable space],
+  typically (as in our case) a subset of $RR$.
 ]
 
 #definition[