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Tassion} +\section{Wahrscheinlichkeitsräume} \input{parts/00_intro.tex} \end{document} diff --git a/semester4/ps/ps-rb/parts/00_intro.tex b/semester4/ps/ps-rb/parts/00_intro.tex index 39f3665..fd62e80 100644 --- a/semester4/ps/ps-rb/parts/00_intro.tex +++ b/semester4/ps/ps-rb/parts/00_intro.tex @@ -30,7 +30,7 @@ $$ \begin{tabular}{ll} (i) & $\P[\emptyset] = 0$ \\ - (ii) & $\displaystyle\overunderset{k}{i=1}{\bigcap} A_i = \emptyset \implies \P\biggl[ \overunderset{k}{i=1}{\bigcup} A_i \biggr] = \sum_{i=1}^{k} \P[A_i]$ \\ + (ii) & $\displaystyle\overunderset{k}{i=1}{\bigcap} A_i = \emptyset \implies \P\Biggl[ \overunderset{k}{i=1}{\bigcup} A_i \Biggr] = \sum_{i=1}^{k} \P[A_i]$ \\ (iii) & $\P[A^\comp] = 1 - \P[A]$ \\ (iv) & $\P[A \cup B] = \P[A] + \P[B] - \P[A \cap B]$ \end{tabular} @@ -58,3 +58,68 @@ $$ (i) & $\F = \mathcal{P}(\Omega)$ \\ (ii) & $\forall A \in \F:\quad \P[A] = \displaystyle\frac{|A|}{|\Omega|}$ \end{tabular} + +\lemma \textbf{Nützliche Ungleichungen} + +\begin{tabular}{lll} + (i) & $A \subseteq B \implies \P[A] \leq \P[B]$ & \subtext{(Monotonie)} \\ + (ii) & $\P \Biggl[ \displaystyle\overunderset{\infty}{i=1}{\bigcup} A_i \Biggr] \leq \sum_{\infty}^{i=1}\P[A_i]$ & \subtext{(Union Bound)} +\end{tabular} + +\subtext{$A_1,A_2,\ldots$ müssen \textit{nicht} disjunkt sein.} + +\lemma \textbf{Stetigkeit} von $\P$ gegen $\infty$\\ +\begin{tabular}{ll} + (i) & $\forall n: A_n \subseteq A_{n+1} \implies \limn \P[A_n] = \P \Biggl[ \displaystyle\overunderset{\infty}{n=1}{\bigcup} A_n \Biggr]$ \\ + (ii) & $\forall n: B_n \supseteq B_{n+1} \implies \limn \P[B_n] = \P \Biggl[ \displaystyle\overunderset{\infty}{n=1}{\bigcap} B_n \Biggr]$ +\end{tabular} + +\subtext{$(A_n), (B_n)$ sind monotone Folgen von Ereignissen} + +\definition \textbf{Bedingte Wahrscheinlichkeit} +$$ + \P\bigl[A\ \big|\ B\bigr] := \frac{\P[A \cap B]}{\P[B]} +$$ +\subtext{$A,B \in \F,\quad \P[B] > 0$} + +\begin{footnotesize} + \lemma $\P\bigl[ A \big| A \bigr] = 1$ + \color{gray} + $\qquad\P[A] > 0$ + \color{black} +\end{footnotesize} + +\lemma \textbf{Totale Wahrscheinlichkeit} +$$ + \forall A \in \F:\quad \P[A] = \sum_{i=1}^{n}\P\bigl[ A \big| B_i \bigr] \cdot \P[B_i] +$$ +\subtext{$B_1,\cdots,B_n$ sind eine Partition von $\Omega$, $\P[B_i] > 0$.} + +\lemma \textbf{Bayes} +$$ + \forall i = 1,\cdots,n:\quad \P\bigl[ B_i \big| A \bigr] = \frac{\P\bigl[ A \big| B_i \bigr] \cdot \P[B_i]}{\sum_{j=1}^{n}\P\bigl[ A \big| B_j \bigr] \cdot \P[B_j]} +$$ +\subtext{$B_1,\cdots,B_n$ sind eine Partition von $\Omega$, $\P[B_i] > 0$, $\P[A] > 0$.} + +\newpage + +\definition \textbf{Unabhängigkeit} +$$ + A, B \text{ unabhängig } \iffdef \P[A \cap B] = \P[A] \cdot \P[B] +$$ + +\lemma \textbf{Äquivalente Aussagen zur Unabhängigkeit} + +\begin{tabular}{lll} + (i) & $\P[A \cap B] = \P[A] \cdot \P[B]$ & \subtext{(Defintion)} \\ + (ii) & $\P[A | B] = \P[A]$ & \subtext{($B$ kein Einfluss auf $A$)} \\ + (iii) & $\P[B | A] = \P[B]$ & \subtext{($A$ kein Einfluss auf $B$)} \\ +\end{tabular} + +\subtext{$A, B \in \F,\quad \P[A], \P[B] > 0$} + +\definition \textbf{Unabhängigkeit} für Ereignissmengen +$$ + (A_i)_{i \in I} \text{ unabhängig } \iffdef \forall J \subseteq I: \P \Biggl[ \underset{j \in J}{\bigcap} A_j \Biggr] = \prod_{j \in J} \P[A_j] +$$ +\subtext{$I$ ist eine Indexmenge. Dies muss für \textit{alle} $J \subseteq I$ (endlich) gelten.} \ No newline at end of file