Add first part of theory for exercise two

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