From ad2213ac1e210444ff62ff07ceabeb864733bd79 Mon Sep 17 00:00:00 2001 From: omicron Date: Tue, 27 May 2025 09:52:17 +0200 Subject: [PATCH] Initial commit, start of notes and generic macros --- .gitignore | 6 ++ Makefile | 26 +++++++ deps/macros.sty | 55 +++++++++++++++ deps/setup.sty | 62 +++++++++++++++++ probability.tex | 182 ++++++++++++++++++++++++++++++++++++++++++++++++ 5 files changed, 331 insertions(+) create mode 100644 .gitignore create mode 100644 Makefile create mode 100644 deps/macros.sty create mode 100644 deps/setup.sty create mode 100644 probability.tex diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..3aca3a1 --- /dev/null +++ b/.gitignore @@ -0,0 +1,6 @@ +*.aux +*.fls +*.fdb_latexmk +*.log +*.pdf +*.out diff --git a/Makefile b/Makefile new file mode 100644 index 0000000..139ef19 --- /dev/null +++ b/Makefile @@ -0,0 +1,26 @@ +DEPS = deps/macros.sty deps/setup.sty +SOURCES = $(wildcard *.tex) +TARGETS = $(SOURCES:.tex=) +WATCH_TARGETS = $(addsuffix -watch,$(TARGETS)) +CHECK_TARGETS = $(addsuffix -check,$(TARGETS)) + +.PHONY: all check clean $(TARGETS) $(WATCH_TARGETS) $(CHECK_TARGETS) + +all: $(TARGETS) + + +check: $(CHECK_TARGETS) + + +$(TARGETS): %: %.tex + latexmk -pdf $< + +$(WATCH_TARGETS): %-watch: %.tex + latexmk -pdf -pvc -interaction=nonstopmode $< + +$(CHECK_TARGETS): %-check: % + @! grep -A1 "Package nag Warning" $*.log + +clean: + latexmk -c + rm -f *.pdf diff --git a/deps/macros.sty b/deps/macros.sty new file mode 100644 index 0000000..8edd931 --- /dev/null +++ b/deps/macros.sty @@ -0,0 +1,55 @@ +\NeedsTeXFormat{LaTeX2e}[1994/06/01] +\ProvidesPackage{omicron/macros}[2025-05-25 omicron/macros package] + +\RequirePackage{mathtools} +\RequirePackage{amssymb} +\RequirePackage{amsthm} + +% create \set and \smid macro to create sets and scaling bar for set builder notation. +\newcommand{\set}[1]{\left\{ #1 \right\}} +\newcommand{\smid}{\,\middle|\,} + +% easier access to blackboard bold +\@ifpackageloaded{dsfont}{% + \let\bb\mathds +}{% + \let\bb\mathbb +} + +% swap default slanted/curly versions of common relations +\let\leqflat\leq +\let\leq\leqslant +\let\geqflat\geq +\let\geq\geqslant +\let\precflateq\preceq +\let\preceq\preccurlyeq +\let\succflateq\succeq +\let\succeq\succcurlyeq + +% swap varepsilon and epsilon +\let\uglyepsilon\epsilon +\let\epsilon\varepsilon +\let\varepsilon\uglyepsilon + +% swap varphi and phi +\let\uglyphi\phi +\let\phi\varphi +\let\varphi\uglyphi + +% scaling abs value +\newcommand{\abs}[1]{\left|#1\right|} + +% scaling parenthesis +\newcommand{\paren}[1]{\left(#1\right)} + +% Probability function +\DeclareMathOperator{\probop}{P} +\newcommand{\prob}[1]{\probop\paren{#1}} + +% Complement superscript operator +\DeclareMathOperator{\complop}{c} +\newcommand{\mycomplement}{{\complop}} +\let\altcomplement\complement +\let\complement\mycomplement + +\endinput diff --git a/deps/setup.sty b/deps/setup.sty new file mode 100644 index 0000000..3ad5ee6 --- /dev/null +++ b/deps/setup.sty @@ -0,0 +1,62 @@ +\NeedsTeXFormat{LaTeX2e}[1994/06/01] +\ProvidesPackage{omicron/setup}[2025-05-25 omicron/setup package] + +% better typesetting +\RequirePackage[T1]{fontenc} +\RequirePackage{lmodern} +\RequirePackage{microtype} +\RequirePackage{bbm} +\RequirePackage{dsfont} + +% math essentials +\RequirePackage{amsmath} +\RequirePackage{amssymb} +\RequirePackage{amsthm} +\RequirePackage{thmtools} +\RequirePackage{mathtools} + +% for title page edits +\RequirePackage{titling} + +% utility +\RequirePackage{enumitem} +\RequirePackage{hyperref} +\RequirePackage{cleveref} +\RequirePackage{todonotes} + +% Helps create better more modern LaTeX +\RequirePackage[l2tabu, orthodox]{nag} + +% lorem ipsum +\RequirePackage{lipsum} + +% No indent style paragraphs +\setlength{\parindent}{0pt} +\setlength{\parskip}{0.5\baselineskip plus 2pt minus 1pt} + +% subtitle macro +\newcommand{\subtitle}[1]{% + \posttitle{% + \par\end{center} + \begin{center} + \parbox{0.7\textwidth}{\centering#1} + \end{center} + \vskip0.5em}% +} + +% Remove the date completely from the title page +\predate{} +\date{} +\postdate{} + +% Essential theorem environments +\declaretheorem[style=definition, name=Definition]{definition} +\declaretheorem[style=definition, sibling=definition, name=Theorem]{theorem} +\declaretheorem[style=definition, sibling=definition, name=Lemma]{lemma} +\declaretheorem[style=definition, sibling=definition, name=Observation]{observation} +\declaretheorem[style=definition, sibling=definition, name=Rules]{rules} + +% default enumeration style +\setlist[enumerate]{label=(\arabic*)} + +\endinput diff --git a/probability.tex b/probability.tex new file mode 100644 index 0000000..71a0b81 --- /dev/null +++ b/probability.tex @@ -0,0 +1,182 @@ +\documentclass{article} + +\usepackage{deps/setup} +\usepackage{deps/macros} +\usepackage{showkeys} + +\title{Probability Notes} +\subtitle{Based on KUL Course Notes for ``Kansrekenen I'' (2018) by Tim Verdonck} +\author{omicron} + +\begin{document} +\maketitle + +\section{Probability Spaces} + +\begin{definition}[Sigma Algebra]\label{def:sigma-algebra} + A collection $\mathcal A$ of subsets of $\Omega$ is called a + \emph{sigma-algebra} (or $\sigma$-algebra) on the universe $\Omega$ if + $\mathcal A$ satisfies the following axioms: + \begin{enumerate} + \item $\Omega \in \mathcal A$, + \item $A \in \mathcal A \implies A^\complement \in \mathcal A$, + \item $\forall n\in\bb N : A_n \in \mathcal A \implies + \paren{\bigcup_{n\in \bb N} A_n} \in \mathcal A$. + \end{enumerate} + We call the pair $(\Omega, \mathcal A)$ a \emph{measurable space} and the + elements of $\mathcal A$ \emph{events}. +\end{definition} + +\begin{observation}\label{obs:has-empty} + Let $(\Omega, \mathcal A)$ be a measurable space, then $\emptyset \in + \mathcal A$. +\end{observation} +\begin{proof} + By \cref{def:sigma-algebra} we have $\Omega \in \mathcal A$ and + $\Omega^\complement = \emptyset \in \mathcal A$. +\end{proof} + +\begin{definition}[Probability Measure]\label{def:probability-measure} + A function $\probop : \mathcal A \to \bb R$ is called a \emph{probability + measure} if it satisfies the following axioms: + \begin{enumerate} + \item $\prob{\Omega} = 1$. + \item $\forall A \in \mathcal A : \prob A \geq 0$. + \item For a family of pairwise disjoint sets $A_1, A_2, \ldots \in + \mathcal A$, + \[ + \prob{\bigcup_{n \in \bb N} A_n} = \sum_{n \in \bb N} + \prob{A_n}. + \] + We call this axiom the axiom of \emph{countable additivity} or + \emph{$\sigma$-additivity}. + \end{enumerate} + The triple $(\Omega, \mathcal A, \probop)$ is called a \emph{probability + space}, comprised of the universe $\Omega$, a $\sigma$-algebra $\mathcal A$ + and a probability measure $\probop$. +\end{definition} +\begin{observation}\label{obs:prob-empty} + Let $(\Omega, \mathcal A, \probop)$ be a probability space, then + $\prob{\emptyset} = 0$. +\end{observation} +\begin{proof} + Note that $\emptyset = \bigcup_{n\in\bb N} \emptyset$ and that the + right-hand side is a union of disjoint sets. By applying the + sigma-additivity axiom we get \[ \prob{\emptyset} = \prob{\bigcup_{n\in\bb + N} \emptyset} = \sum_{n\in\bb N} \prob{\emptyset}. \] Since $P$ takes real + values we know the series must converge. This can only happen if + $P(\emptyset) = 0$. +\end{proof} + +\begin{definition}[Monotonous sequence of sets]\label{def:monotonous-sets} + A sequence of sets $\paren{A_n}_{n\in\bb N_0}$ is said to be + \emph{increasing} if $A_n \subseteq A_{n+1}$ for + every $n \in \bb N_0$. Similarly, a sequence is called \emph{decreasing} + if $A_n \supseteq A_{n+1}$ for every $n \in \bb + N_0$. A sequence is called \emph{monotonous} if it is either increasing or + decreasing. For such sequences we define + \[ + \lim_{n\to\infty} A_n = \begin{cases} + \bigcup_{n=1}^{\infty} A_n & \text{if $A_n$ is increasing}, \\ + \bigcap_{n=1}^{\infty} A_n & \text{if $A_n$ is decreasing}. \\ + \end{cases} + \] +\end{definition} + +\begin{theorem} + Let $(\Omega, \mathcal A, \probop)$ be a probability space. + \begin{enumerate} + \item \emph{Finite additivity} for a pairwise disjoint family of sets + $\set{A_n \in \mathcal A \smid n \in \set{1, \dots, N}}$ + \[ + \prob{\bigcup_{n=1}^{N} A_n} = \sum_{n=1}^{N} \prob{A_n}. + \] + \item $\forall A \in \mathcal A : \prob{A^\complement} = 1 - \prob{A}$. + \item For a monotonous sequence $\paren{A_n}_{n\in\bb N_0}$ + \[ + \prob{\lim_{n\to\infty} A_n} = \lim_{n\to\infty} \prob{A_n} + \] + \end{enumerate} +\end{theorem} +\begin{proof} + \begin{enumerate} + \item We start by showing finite additivity as defined above holds. Consider the + finite family of sets defined above, we define a related infinite family of + sets as follows + \[ + B_n = \begin{cases} + A_n & \text{if $n \leq N$}, \\ + \emptyset & \text{otherwise}. + \end{cases} + \] + Note that since we only added empty sets this new family is also pairwise + disjoint and that the countable union over $B_n$ is equal to the finite + union over $A_n$. This leads to + \begin{align*} + \prob{\bigcup_{n=1}^{N} A_n} + &= \prob{\bigcup_{n=1}^{\infty} B_n}. \\ + \intertext{Next we apply countable additivity by \cref{def:probability-measure} and get} + &= \sum_{n=1}^{\infty} \prob{B_n} \\ + &= \sum_{n=1}^{N} \prob{A_n} + \sum_{n=N+1}^{\infty} \prob{\emptyset}. + \intertext{In \cref{obs:prob-empty} we concluded $\prob{\emptyset} = 0$ so} + &= \sum_{n=1}^{N} \prob{A_n}. + \end{align*} + This concludes the proof of finite additivity. + + \item Let $A \in \mathcal A$ be arbitrary. By + \cref{def:probability-measure}, finite additivity and the definition of + complement we get + \[ + 1 = \prob{\Omega} = \prob{A \cup A^\complement} = \prob{A} + \prob{A^\complement}. + \] + Subtracting $\prob{A}$ from both sides gives us + \[ + \prob{A^\complement} = 1 - \prob{A}. + \] + This concludes the proof of the second point. + \item Let $(A_n)_{n\in\bb N_0}$ be a monotonous sequence in $\mathcal A$. + We first consider the case where $(A_n)$ is increasing. We define a + related sequence as follows + \[ + B_n = \begin{cases} + A_1 & \text{if $n=1$,} \\ + A_n \setminus A_{n-1} & \text{otherwise.} + \end{cases} + \] + Note that $(B_n)$ is pairwise disjoint and that for any $n \in \bb N_0$ + \[ + \bigcup_{i=1}^n A_i = \bigcup_{i=1}^n B_i = A_n + \] + By using the properties of this related sequence and by applying sigma + additivity we get + \begin{align*} + \prob{\lim_{n\to\infty} A_n} + &= \prob{\bigcup_{i=1}^\infty A_i} \\ + &= \prob{\bigcup_{i=1}^\infty B_i} \\ + &= \sum_{i=1}^\infty \prob{B_i}. \\ + \intertext{Next we use the definition of a series and apply finite + additivity (in reverse) as follows} + &= \lim_{n\to\infty} \sum_{i=1}^n \prob{B_i} \\ + &= \lim_{n\to\infty} \prob{\bigcup_{i=1}^n B_i} \\ + &= \lim_{n\to\infty} \prob{A_n}. + \end{align*} + This concludes of the proof when $(A_n)$ is increasing. Next assume + that $(A_n)$ is decreasing. Note that the complement sequence + $(A_n^\complement)$ is an increasing sequence. Combined with De + Morgan's laws and the earlier formula for the probability of a + complement we get + \begin{align*} + \prob{\lim_{n\to\infty} A_n} + &= \prob{\bigcap_{i=1}^\infty A_i} \\ + &= 1 - \prob{\bigcup_{i=1}^\infty A_i^\complement} \\ + &= 1 - \prob{\lim_{n\to\infty} A_n^\complement}. \\ + \intertext{since the complement sequence is an increasing sequence + we can apply the result from before and we get} + &= 1 - \lim_{n\to\infty} \prob{A_n^\complement} \\ + &= 1 - \lim_{n\to\infty} 1 - \prob{A_n} \\ + &= \lim_{n\to\infty} \prob{A_n}. + \end{align*} + This concludes the proof for the final case. \qedhere + \end{enumerate} +\end{proof} +\end{document}