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@@ -0,0 +1,60 @@ +\definition \textbf{Grundraum} $\Omega\qquad$ \textbf{Elementarereignis} $\omega \in \Omega$ + +\definition \textbf{$\sigma$-Algebra} $\quad \mathcal{F} \subseteq \mathcal{P}(\Omega)\qquad$ \textbf{Ereignis} $A \in \mathcal{F}$ + +\begin{tabular}{lll} + (i) & $\Omega \in \mathcal{F}$ \\ + (ii) & $A \in \mathcal{F}$ & $\implies A^\comp \in \mathcal{F}$ \\ + (iii) & $A_1,\cdots,A_n \in \mathcal{F}$ & $\implies \displaystyle\underset{i \leq n}{\bigcup} A_i \in \mathcal{F}$ +\end{tabular} + +\lemma \textbf{Abgeschlossenheit} der $\sigma$-Algebra $\F$ + +\begin{tabular}{ll} + (i) & $\emptyset \in \F$ \\ + (ii) & $A_1,\cdots,A_n \in \F \implies \displaystyle\overunderset{\infty}{i=1}{\bigcap} A_i \in \F$ \\ + (iii) & $A, B \in \F \implies A \cup B \in \F$ \\ + (iv) & $A, B \in \F \implies A \cap B \in \F$ +\end{tabular} + +\definition \textbf{Wahrscheinlichkeitsmass} auf $(\Omega, \F): \P$ +$$ + \P: \F \to [0,1] \qquad \text{s.d.} \qquad A \mapsto \P[A] +$$ +\begin{tabular}{ll} + (i) & $\P[\Omega] = 1$ \\ + (ii) & $\P[A] = \displaystyle\sum_{i=1}^{\infty} A_i \iff A = \overunderset{\infty}{i=1}{\bigcup}A_i \quad \text{s.d.} \quad \overunderset{\infty}{i=1}{\bigcap}A_i = \emptyset$ +\end{tabular} + +\lemma \textbf{Eigenschaften} von $\P$ + +\begin{tabular}{ll} + (i) & $\P[\emptyset] = 0$ \\ + (ii) & $\displaystyle\overunderset{k}{i=1}{\bigcap} A_i = \emptyset \implies \P\biggl[ \overunderset{k}{i=1}{\bigcup} A_i \biggr] = \sum_{i=1}^{k} \P[A_i]$ \\ + (iii) & $\P[A^\comp] = 1 - \P[A]$ \\ + (iv) & $\P[A \cup B] = \P[A] + \P[B] - \P[A \cap B]$ +\end{tabular} + +\definition \textbf{Wahrscheinlichkeitsraum} $(\Omega, \F, \P)$ +\subtext{$A \in \mathcal{F}, \quad \omega \in \Omega$} + +\begin{tabular}{lll} + $A$ tritt ein &$\iffdef$& $\omega \in A$ \\ + $A$ tritt nicht ein &$\iffdef$& $\omega \notin A$ +\end{tabular} +\footnotesize +\begin{tabular}{ll} + (i) & $\emptyset$ tritt nie ein \\ + (ii) & $\Omega$ tritt immer ein +\end{tabular} +\normalsize + +\newpage + +\definition \textbf{Laplace Modell} $(\Omega, \F, \P)$ +\subtext{$\Omega$ endlich.} + +\begin{tabular}{ll} + (i) & $\F = \mathcal{P}(\Omega)$ \\ + (ii) & $\forall A \in \F:\quad \P[A] = \displaystyle\frac{|A|}{|\Omega|}$ +\end{tabular} diff --git a/semester4/ps/ps-rb/util/helpers.tex b/semester4/ps/ps-rb/util/helpers.tex index 281b160..60dac48 100644 --- a/semester4/ps/ps-rb/util/helpers.tex +++ b/semester4/ps/ps-rb/util/helpers.tex @@ -17,7 +17,7 @@ base={#1} } -% Math helpers +% Math helpers, General \def\limxo{\lim_{x\to 0}} \def\limxi{\lim_{x\to\infty}} \def\limxn{\lim_{x\to-\infty}} @@ -38,6 +38,7 @@ \def \cgeq{\succcurlyeq} \def \cleq{\preccurlyeq} +% Math helpers, Analysis \def \limn{\lim\limits_{n \to \infty}} \def \limi{\liminf\limits_{n \to \infty}} \def \lims{\limsup\limits_{n \to \infty}} @@ -53,6 +54,12 @@ \def \ssdfd#1{\partial_#1f} \def \dd#1#2{\frac{\partial_#1}{\partial x_#2}} % to replace f +% Math helpers, Probability +\def \comp{\text{C}} + +\def \P{\mathbb{P}} +\def \F{\mathcal{F}} + % Titles \def \definition{\colorbox{lightgray}{Def} } \def \notation{\colorbox{lightgray}{Notation} }