From 08a1324efed22ba94e59faff9ebec79c247b4c1f Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Sat, 21 Feb 2026 19:54:49 +0100 Subject: [PATCH] [PS] Continue introduction --- .../parts/00_basics/00_probability-space.tex | 24 ++++++++++++++++-- .../probability-and-statistics-cheatsheet.pdf | Bin 206998 -> 220427 bytes .../probability-and-statistics-cheatsheet.tex | 1 + 3 files changed, 23 insertions(+), 2 deletions(-) diff --git a/semester4/ps/ps-jh/parts/00_basics/00_probability-space.tex b/semester4/ps/ps-jh/parts/00_basics/00_probability-space.tex index c04ece9..22d8f0c 100644 --- a/semester4/ps/ps-jh/parts/00_basics/00_probability-space.tex +++ b/semester4/ps/ps-jh/parts/00_basics/00_probability-space.tex @@ -8,16 +8,36 @@ \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 ($\Omega = \{ 1, 2, 3, 4, 5, 6 \}$) +\shortexample $\sigma$-Algebren beim einmal. 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: +Keine $\sigma$-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} + +\subsubsection{Wahrscheinlichkeitsmass} +\shortdefinition[W.M] $\P : \cF \rightarrow [0, 1]$ mit $A \mapsto \P[A]$, notiert $(\Omega, \cF)$ und falls folgende Eigenschaften gelten +\begin{enumerate}[label=E\arabic*] + \item $\P[\Omega] = 1$ + \item ($\sigma$-\bi{Additivität}) $\P[A] = \sum_{i = 1}^{\infty} \P[A_i]$,\\ + falls $A = \bigcup_{i = 1}^\infty A_i$ \textit{(disjunkte Vereinigung)} +\end{enumerate} +\shortexample Wieder mit Würfeln und $\cF = \cP(\Omega)$, sind W.M: +\begin{itemize} + \item Abbildung $\forall A \in \cF \quad \P[A] = \frac{|A|}{6}$ + \item Abbildung $\forall A \in \cF \quad \P[A] = \sum_{i \in A} p_i$ + ($p_i$ dabei prob. Zahl $i$ würfeln; $p_i = \frac{1}{6} \forall i \in \Omega$ ist für fairen Würfel) +\end{itemize} + + +\subsubsection{Wahrscheinlichkeitsraum} +\shortdefinition[W.R] ein Tripel $(\Omega, \cF, \P)$ + +\shortterm $A$ Ereignis, \bi{tritt (nicht) ein} (für $\omega$), if $\omega \in (\notin) A$ diff --git a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf index 6c72d33f53833bb6333e1a622bf8a2b116a12675..17009099f05e7d8aa67d4e821d243fcbb1575216 100644 GIT binary patch delta 170805 zcmYhBV|$)$*k>1&U? zV54v3PFj&nI)Q0d%8<}TO(4=U)hq;u7E>mr`sMDLn{$Y}cZ!H3o#_%^>0m0 zYm1?>sR{)!FRtb~Uw-(QSHxq2T!=Pn0ynDMy!%?`W)U(yMw-1&EAjotBm7gP|M#W( z^*^ucn><0{OuVFzqHzZdM2tv8Zd_-^cqKl59R-DLoHgEV_AI5E#yIz5Hq{U@8MbHN z1R{lP`N-GR=_VMLP+Gb}n)KA*${U*B^&-&_aIlgXfU zF9|FO-M`W7NSezRytyhDtWdwxbVmDWSVj}m(J3elixX5hU#=Zy%Gl{l{n|4eD*u(f z8k!Hrk`Lip%|!f&7D9i_LJLVAwLen4&$33ciJ1Cw>U$YtgwVQ81uK*?wuh_K>N!|8 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\input{parts/00_basics/00_probability-space.tex} +% \input{parts/00_basics/} \end{document}

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