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Andreas Tsouchlos 2026-01-16 04:23:32 +01:00
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src/2026-01-16/presentation.tex
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@ -291,64 +291,6 @@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Theorie Wiederholung} \subsection{Theorie Wiederholung}
\begin{frame}
\frametitle{Unabhängigkeit \& Korrelation}
\vspace*{-10mm}
\begin{itemize}
\item Unabhängige ZV (stetig)
\begin{columns}
\column{\kitthreecolumns}
\begin{align*}
X,Y \text{ unabhängig}
\hspace{5mm} \Leftrightarrow \hspace{5mm}
f_{X,Y}(x,y) = f_X(x)f_Y(y)
\end{align*}
\column{\kitthreecolumns}
\begin{lightgrayhighlightbox}
Erinnerung: Unabhängige Ereignisse
\begin{align*}
X,Y \text{ \normalfont unabhängig}
\hspace{5mm} \Leftrightarrow \hspace{5mm}
P(AB) = P(A)P(B)
\end{align*}
\vspace*{-13mm}
\end{lightgrayhighlightbox}
\end{columns}
\pause
\item Kovarianz
\begin{columns}
\column{\kitthreecolumns}
\begin{align*}
\text{cov}(X,Y) &= E\bigg( \big(X - E(X)\big) \big(Y
- E(Y)\big) \bigg) \\
&= E(XY) - E(X)E(Y)
\end{align*}
\column{\kitthreecolumns}
\begin{lightgrayhighlightbox}
Erinnerung: Varianz
\begin{align*}
V(X) = E\big( \left(X - E(X)\right)^2 \big) = E(X^2) - E^2(X)
\end{align*}
\vspace*{-13mm}
\end{lightgrayhighlightbox}
\end{columns}
\item Korrelation
\begin{align*}
E(XY)
\end{align*}
\pause
\item Korrelationskoeffizient
\begin{align*}
\rho_{XY} = \frac{\text{cov}(X,Y)}{\sqrt{V(X)V(Y)}}
\hspace{25mm} \rho_{XY} = 0
\hspace{2mm}\Leftrightarrow\hspace{2mm}
E(XY) = E(X)E(Y)
\end{align*}
\end{itemize}
\end{frame}
\begin{frame} \begin{frame}
\frametitle{Mehrdimensionale Zufallsvariablen} \frametitle{Mehrdimensionale Zufallsvariablen}
@ -457,7 +399,65 @@
\end{frame} \end{frame}
\begin{frame} \begin{frame}
\frametitle{Unabhängigkeit vs. Korrelation} \frametitle{Unabhängigkeit \& Korrelation I}
\vspace*{-10mm}
\begin{itemize}
\item Unabhängige ZV (stetig)
\begin{columns}
\column{\kitthreecolumns}
\begin{align*}
X,Y \text{ unabhängig}
\hspace{5mm} \Leftrightarrow \hspace{5mm}
f_{X,Y}(x,y) = f_X(x)f_Y(y)
\end{align*}
\column{\kitthreecolumns}
\begin{lightgrayhighlightbox}
Erinnerung: Unabhängige Ereignisse
\begin{align*}
X,Y \text{ \normalfont unabhängig}
\hspace{5mm} \Leftrightarrow \hspace{5mm}
P(AB) = P(A)P(B)
\end{align*}
\vspace*{-13mm}
\end{lightgrayhighlightbox}
\end{columns}
\pause
\item Kovarianz
\begin{columns}
\column{\kitthreecolumns}
\begin{align*}
\text{cov}(X,Y) &= E\bigg( \big(X - E(X)\big) \big(Y
- E(Y)\big) \bigg) \\
&= E(XY) - E(X)E(Y)
\end{align*}
\column{\kitthreecolumns}
\begin{lightgrayhighlightbox}
Erinnerung: Varianz
\begin{align*}
V(X) = E\big( \left(X - E(X)\right)^2 \big) = E(X^2) - E^2(X)
\end{align*}
\vspace*{-13mm}
\end{lightgrayhighlightbox}
\end{columns}
\item Korrelation
\begin{align*}
E(XY)
\end{align*}
\pause
\item Korrelationskoeffizient
\begin{align*}
\rho_{XY} = \frac{\text{cov}(X,Y)}{\sqrt{V(X)V(Y)}}
\hspace{25mm} \rho_{XY} = 0
\hspace{2mm}\Leftrightarrow\hspace{2mm}
E(XY) = E(X)E(Y)
\end{align*}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Unabhängigkeit \& Korrelation II}
\vspace*{-15mm} \vspace*{-15mm}