[PS] Start summarizing

README.md

@@ -34,7 +34,7 @@ If you find mistakes, please open an issue or fix them yourself and open a PR.
 
 ### Semester 4
 #### Probability & Statistics
-- [Probability & Statistics Cheat-Sheet (TBD)](./semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf) Author: Janis Hutz
+- [Probability & Statistics Cheat-Sheet (DE)](./semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf) Author: Janis Hutz
 - [Probability & Statistics Cheat-Sheet (TBD)](./semester4/ps/ps-rb/main.pdf) Author: Robin Bacher
 #### Other Courses
 - [Formal Methods & Functional Programming Summary (EN)](./semester4/fmfp/formal-methods-functional-programming-summary.pdf)

semester4/ps/ps-jh/parts/00_basics/00_intro.tex

@@ -1 +0,0 @@
-\inline{definition}

semester4/ps/ps-jh/parts/00_basics/00_probability-space.tex

@@ -0,0 +1,23 @@
+\subsection{Wahrscheinlichkeitsräume}
+\shortterm $\Omega$ \bi{Grundraum}, $\omega \in \Omega$ \bi{Elementarereignis}
+
+\shortdefinition[Sigma-Algebra] $\cF \subseteq \cP(\Omega)$ ist $\sigma$-Algebra, falls:
+\begin{enumerate}[label=E\arabic*.]
+    \item $\Omega \subseteq \cF$
+    \item $A \in \cF \Rightarrow A^C \in \cF$ ($A$ Ereignis $\Rightarrow$ nicht $A$ auch)
+    \item $A_1, A_2, \ldots \in \cF \Rightarrow \bigcup_{i = 1}^\infty A_i \in \cF$\\
+        ($A_1, \ldots$ Ereignisse $\Rightarrow$ $A_1$ oder $A_2$ oder \dots ein Ereignis)
+\end{enumerate}
+\shortexample $\omega$-Algebren beim einmaligen Würfeln ($\Omega = \{ 1, 2, 3, 4, 5, 6 \}$)
+\begin{itemize}
+    \item $\cF = \{ \emptyset, \{ 1, 2, 3, 4, 5, 6 \} \}$
+    \item $\cF = \cP(\Omega)$, dabei $|\cF| = 64$
+    \item $\cF = \{ \emptyset, \{ 1, 2 \}, \{ 3, 4, 5, 6 \}, \Omega \}$
+\end{itemize}
+Keine $\omega$-Algebren sind bspw:
+\begin{itemize}
+    \item $\cF = \{ \Omega \}$ (Komplementärereignis $\emptyset$ fehlt, E2 verletzt)
+    \item $\cF = \{ \emptyset, \{ 1, 2, 3 \}, \{ 4, 5, 6 \}, \{ 1 \}, \{ 2, 3, 4, 5, 6 \}, \Omega \}$\\
+        (E3 verletzt, da bspw $\{ 4, 5, 6 \} \cup \{ 1 \} \notin \cF$)
+\end{itemize}
+

semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex

@@ -1,13 +1,22 @@
 \documentclass{article}
 
+\PassOptionsToPackage{skip=0pt}{parskip}
 \input{~/projects/latex/janishutz-helpers.tex}
+\usepackage{lmodern}
+\setFontType{sans}
 
-\setup{Probability and Statistics}
+\setupCheatSheet{Wahrscheinlichkeit und Statistik}
+\loadGerman
+
+\setsubsectionnumbering{section}
+\renewcommand{\examplenumbering}{off}
 
 \begin{document}
 \startDocument
+\noverticalspacing
 
-\input{parts/00_basics/00_intro.tex}
+\section{Basics}
+\input{parts/00_basics/00_probability-space.tex}
 
 
 \end{document}