auto-update(nvim): 2025-02-19 18:00:46

documents/by-course/pstat-120a/course-notes/main.typ

@@ -907,8 +907,27 @@ getting $P(B | A)$ from $P(A | B)$.
 
 == Random variables, discrete random variables
 
-Recall that a random variable $X$ is a function $X : Omega -> RR$ that gives
-the probability of an event $omega in Omega$. The _probability distribution_ of
+First, some brief exposition on random variables. Quixotically, a random
+variable is actually a function.
+
+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 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[
+  The *state space* or *support* of a random variable $X$ is all of the values $X$ can take.
+]
+
+#example[
+  Let $X$ be a random variable that takes on the values ${0,1,2,3}$. Then the
+  state space of $X$ is the set ${0,1,2,3}$.
+]
+
 $X$ gives its important probabilistic information. The probability distribution
 is a description of the probabilities $P(X in B)$ for subsets $B in RR$. We
 describe the probability density function and the cumulative distribution