tut3: Add theory for exercise 2
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@ -206,7 +206,7 @@
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\end{lightgrayhighlightbox}
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\end{columns}
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\pause
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\item Einige Kenngrößen von Verteilungen
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\item Kenngrößen von Verteilungen
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\vspace*{2mm}
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\begin{columns}[t]
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\column{\kittwocolumns}
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@ -310,7 +310,7 @@
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\begin{frame}
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\frametitle{Zusammenfassung}
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\begin{columns}
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\begin{columns}[t]
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\column{\kittwocolumns}
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\begin{greenblock}{Verteilungsfunktion (diskret)}
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\vspace*{-6mm}
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@ -557,64 +557,116 @@
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Theorie Wiederholung}
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% \begin{frame}
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% \frametitle{Zusätzliche Bedingungen und Unabhängigkeit}
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%
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% \begin{itemize}
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% \item Erweiterte Definition der bedingten Wahrscheinlichkeit
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% \begin{gather*}
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% P(A\vert BC) = \frac{P(AB\vert C)}{P(B\vert C)}
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% \end{gather*}
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% \item Satz von Bayes mit zusätzlichen Bedingungen
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% \begin{gather*}
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% P(A\vert BC) = \frac{P(B\vert AC) P(A\vert C)}{P(B\vert C)}
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% \end{gather*}
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% \pause
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% \item Unabhängigkeit
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% \begin{gather*}
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% A,B \text{ Unabhängig} \hspace{5mm}
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% \Leftrightarrow\hspace{5mm} P(AB) = P(A) P(B)
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% \hspace{5mm} \Leftrightarrow \hspace{5mm} P(A\vert B) = P(A)
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% \end{gather*}
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% \end{itemize}
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% \end{frame}
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%
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% \begin{frame}
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% \frametitle{Zusammenfassung}
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%
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% \begin{columns}
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% \column{\kitthreecolumns}
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% \begin{greenblock}{Bedingte Wahrscheinlichkeit}
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% \vspace*{-6mm}
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% \begin{gather*}
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% P(A\vert B) = \frac{P(AB)}{P(B)}
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% \end{gather*}
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% \end{greenblock}
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% \column{\kitthreecolumns}
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% \begin{greenblock}{Formel von Bayes}
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% \vspace*{-6mm}
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% \begin{gather*}
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% P(A\vert B) = \frac{P(B\vert A) P(A)}{P(B)}
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% \end{gather*}
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% \end{greenblock}
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% \end{columns}
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% \begin{columns}
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% \column{\kitthreecolumns}
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% \begin{greenblock}{Satz der totalen Wahrscheinlichkeit}
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% \vspace*{-6mm}
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% \begin{gather*}
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% P(B) = \sum_{n} P(B\vert A_n)P(A_n)
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% \end{gather*}
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% \end{greenblock}
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% \column{\kitthreecolumns}
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% \begin{greenblock}{Unabhängigkeit von Ereignissen}
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% \vspace*{-6mm}
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% \begin{gather*}
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% P(AB) = P(A) P(B)
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% \end{gather*}
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% \end{greenblock}
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% \end{columns}
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% \end{frame}
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\begin{frame}
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\frametitle{Weitere Kenngrößen von Verteilungen}
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\vspace*{-10mm}
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\vspace*{10mm}
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\begin{columns}[t]
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\column{\kitthreecolumns}
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\centering
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\textbf{$k$-tes Moment}
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\begin{gather*}
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E(X^k) = \sum_{n=1}^{\infty} x_n^k P(X=x_n)
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\end{gather*}%
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\column{\kitthreecolumns}
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\centering
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\textbf{$k$-tes zentrales Moment}
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\begin{gather*}
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E\left( \left(X - E(X)\right)^k \right) =
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\sum_{n=1}^{\infty} \left(x_n - E(X)\right)^k P(X=x_n)
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\end{gather*}%
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\end{columns}
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\vspace*{20mm}
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\pause
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\begin{columns}[t]
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\column{\kitthreecolumns}
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\centering
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\textbf{Charakteristische Funktion (diskret)}
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\begin{gather*}
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\phi_X(s) = E(e^{jsX}) = \sum_{n=1}^{\infty}
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e^{jsx_n} P(X=x_n)\\[5mm]
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E(X^k) = \frac{\phi_X^{(k)}(0)}{j^k}
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\end{gather*}
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\column{\kitthreecolumns}
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\centering
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\textbf{Erzeugende Funktion}
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\begin{gather*}
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\text{Voraussetzung:} \hspace{5mm} x \in \mathbb{N}_0\\[5mm]
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\psi(z) = E(z^x) = \sum_{n=1}^{\infty} z^n P(x=n)\\[5mm]
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P(X=n) = \frac{\psi_X^{(n)}(0)}{n!}
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\end{gather*}
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\end{columns}
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\end{frame}
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\begin{frame}
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\frametitle{Zusammenfassung}
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\vspace*{-16mm}
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\begin{columns}[t]
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\column{\kittwocolumns}
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\begin{greenblock}{Verteilungsfunktion (diskret)}
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\vspace*{-6mm}
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\begin{gather*}
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F_X(x) = P(X \le x) = \sum_{n:x_n < x} P_X(x_n)
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\end{gather*}
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\end{greenblock}
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\column{\kittwocolumns}
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\begin{greenblock}{Varianz}
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\vspace*{-6mm}
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\begin{gather*}
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V(X) = E\left(\left(X - E(X)\right)^2\right)
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\end{gather*}%
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\vspace*{-8mm}
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\begin{align*}
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V(X) &= E(X^2) - \left(E(X)\right)^2\\
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V(aX) &= a^2 V(x)\\
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V(X+b) &= V(X)
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\end{align*}
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\end{greenblock}
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\column{\kittwocolumns}
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\begin{greenblock}{$p$-Quantil}
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\vspace*{-6mm}
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\begin{gather*}
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x_p = \text{inf}\mleft\{ x\in \mathbb{R} : P(X
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\le x) \ge p \mright\}
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\end{gather*}
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\vspace*{-8mm}
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\begin{gather*}
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p=0.5 \hspace{5mm} \rightarrow \hspace{5mm} x_p
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\equiv \text{``Median''}
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\end{gather*}
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\end{greenblock}
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\end{columns}
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\begin{columns}[t]
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\column{\kittwocolumns}
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\begin{greenblock}{$k$-tes Moment}
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\vspace*{-6mm}
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\begin{gather*}
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E(X^k) = \sum_{n=1}^{\infty} x_n^k P(X=x_n)
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\end{gather*}%
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\end{greenblock}
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\column{\kittwocolumns}
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\begin{greenblock}{Charakt. Funktion (diskret)}
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\vspace*{-6mm}
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\begin{gather*}
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\phi_X(s) = \sum_{n=1}^{\infty}
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e^{jsx_n} P(X=x_n)\\[5mm]
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E(X^k) = \frac{\phi_X^{(k)}(0)}{j^k}
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\end{gather*}
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\end{greenblock}
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\column{\kittwocolumns}
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\begin{greenblock}{Erzeugende Funktion}
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\vspace*{-6mm}
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\begin{gather*}
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\psi(z) = \sum_{n=1}^{\infty} z^n P(x=n)\\[5mm]
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P(X=n) = \frac{\psi_X^{(n)}(0)}{n!}
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\end{gather*}
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\end{greenblock}
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\end{columns}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Aufgabe}
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