diff --git a/src/2026-01-16/presentation.tex b/src/2026-01-16/presentation.tex index af7ff08..3bd1187 100644 --- a/src/2026-01-16/presentation.tex +++ b/src/2026-01-16/presentation.tex @@ -291,64 +291,6 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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} \frametitle{Mehrdimensionale Zufallsvariablen} @@ -457,7 +399,65 @@ \end{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}