diff --git a/src/2026-02-13/presentation.tex b/src/2026-02-13/presentation.tex index c4db74f..5cc6d51 100644 --- a/src/2026-02-13/presentation.tex +++ b/src/2026-02-13/presentation.tex @@ -886,7 +886,112 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Theorie Wiederholung} -% TODO: Add slides +\begin{frame} + \frametitle{Empirische Kenngrößen I} + + \vspace*{-10mm} + + \begin{itemize} + \item Empirischer Erwartungswert + \end{itemize} + + \begin{minipage}{0.47\textwidth} + \centering + \begin{align*} + \overline{x} = \frac{1}{N} \nsum_{i=1}^{N} x_i + \end{align*} + \end{minipage}% + \begin{minipage}{0.53\textwidth} + \centering + \begin{lightgrayhighlightbox} + \vspace*{-3mm} + Erinnerung: Erwartungswert (diskret) + \begin{align*} + E(X) = \nsum_{n=1}^{\infty} x_n P(X=x_n) + \end{align*} + \vspace*{-10mm} + \end{lightgrayhighlightbox} + \end{minipage}% + + \vspace*{10mm} + + \pause + \begin{itemize} + \item Empirische Varianz + \end{itemize} + + \begin{minipage}{0.47\textwidth} + \centering + \begin{align*} + s^2 = \frac{1}{N-1} \nsum_{i=1}^{N} (x_i - \overline{x})^2 + \end{align*} + \end{minipage}% + \begin{minipage}{0.53\textwidth} + \centering + \begin{lightgrayhighlightbox} + \vspace*{-3mm} + Erinnerung: Varianz (diskret) + \begin{align*} + V(X) = E\left( \left( X - E(X) \right)^2 + \right) = \nsum_{n=1}^{\infty} \left( x_n - + E(X) \right)^2 P(X=x_n) + \end{align*} + \vspace*{-10mm} + \end{lightgrayhighlightbox} + \end{minipage} +\end{frame} + +\begin{frame} + \frametitle{Empirische Kenngrößen II} + + \vspace*{-10mm} + + \begin{itemize} + \item Geordnete Stichprobe + \begin{align*} + \begin{pmatrix} + x_1 & \cdots & x_N + \end{pmatrix} + \hspace{10mm} \rightarrow \hspace{10mm} + \begin{pmatrix} + x_{(1)} & \cdots & x_{(N)} + \end{pmatrix}, \hspace{5mm} x_{(1)} \le \cdots \le x_{(N)} + \end{align*} + \pause + \item Empirischer Median + \begin{align*} + x_{1/2} = + \begin{cases} + x_{\left( \frac{N+1}{2} \right)}, & N \text{ + ungerade} \\[3mm] + \frac{1}{2} \left( x_{\left( \frac{N}{2} \right)} + + x_{\left( \frac{N}{2} +1 \right)} \right), & N + \text{ gerade} + \end{cases} + \end{align*} + \pause + \item $p$-Quantil + \begin{align*} + x_{p} = + \begin{cases} + x_{\left( \lfloor Np + 1 \rfloor \right)}, & Np + \notin \mathbb{N} \\[3mm] + \frac{1}{2} \left( x_{\left( Np \right)} + + x_{\left( Np + 1 \right)} \right), & Np \in \mathbb{N} + \end{cases} + \end{align*} + \item Quartilsabstand + \begin{align*} + x_{3/4} - x_{1/4} + \end{align*} + \end{itemize} +\end{frame} + +\begin{frame} + \frametitle{Boxplots} + + % TODO: Create slide +\end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Aufgabe} @@ -959,7 +1064,6 @@ Ergebnisse einen Vorteil des Quartilsabstands gegenüber der Varianz als Maß für die Streuung. \end{enumerate} - \end{frame} % TODO: Boxplot erklären