From 83e16e7aa093a1671b1987374f71336bf7611d95 Mon Sep 17 00:00:00 2001 From: Youwen Wu Date: Wed, 19 Feb 2025 18:00:46 -0800 Subject: [PATCH] auto-update(nvim): 2025-02-19 18:00:46 --- .../pstat-120a/course-notes/main.typ | 23 +++++++++++++++++-- 1 file changed, 21 insertions(+), 2 deletions(-) diff --git a/documents/by-course/pstat-120a/course-notes/main.typ b/documents/by-course/pstat-120a/course-notes/main.typ index 30e3710..91cf0e4 100644 --- a/documents/by-course/pstat-120a/course-notes/main.typ +++ b/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