From 530c70cc828589fc688f2acd2690aac5c52a6810 Mon Sep 17 00:00:00 2001 From: Youwen Wu Date: Sun, 16 Feb 2025 13:59:19 -0800 Subject: [PATCH] chore: clarify random variable --- src/posts/2025-02-16-probability-distributions.md | 9 ++++++--- typst/2025-02-16-probability-distributions.typ | 8 +++++--- 2 files changed, 11 insertions(+), 6 deletions(-) diff --git a/src/posts/2025-02-16-probability-distributions.md b/src/posts/2025-02-16-probability-distributions.md index ebbf22d..fcb9742 100644 --- a/src/posts/2025-02-16-probability-distributions.md +++ b/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. * diff --git a/typst/2025-02-16-probability-distributions.typ b/typst/2025-02-16-probability-distributions.typ index 3277161..0810098 100644 --- a/typst/2025-02-16-probability-distributions.typ +++ b/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[