Add tutorial 7 corrections and fix handout version

src/2026-02-13/presentation.tex

@@ -129,6 +129,123 @@
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{Theorie Wiederholung}
 
+\ifdefined\ishandout
+\begin{frame}
+    \frametitle{Wahrscheinlichkeitstheorie und Statistik}
+
+    \vspace*{-5mm}
+
+    \begin{itemize}
+        \item Einfache Stichprobe
+            \begin{gather*}
+                X_1, \ldots, X_N
+                \hspace{2mm}\overbrace{\text{unabhängig und haben
+                dieselbe Verteilung}}^{\text{``iid.''}}
+                \hspace*{5mm} \rightarrow\hspace*{5mm}
+                \bm{X} :=
+                \begin{pmatrix}
+                    X_1 \\
+                    \vdots \\
+                    X_N
+                \end{pmatrix}
+            \end{gather*}
+    \end{itemize}
+
+    \begin{figure}[H]
+        \centering
+
+        \begin{subfigure}{0.5\textwidth}
+            \centering
+
+            \begin{itemize}
+                \item Wahrscheinlichkeitstheorie
+            \end{itemize}
+            \vspace*{2mm}
+            \begin{tikzpicture}
+                \node[
+                    rectangle,
+                    minimum width=7cm, minimum height=4cm,
+                    line width=1pt,
+                    draw=kit-blue, fill=kit-blue!20,
+                ] (model) {
+                    $\bm{X} =
+                    \begin{pmatrix}
+                        X_1 \\
+                        \vdots \\
+                        X_N
+                    \end{pmatrix}\sim P_{\bm{X}}$
+                };
+
+                \node[right=of model] (x) {
+                    $\bm{x} =
+                    \begin{pmatrix}
+                        x_1 \\
+                        \vdots \\
+                        x_N
+                    \end{pmatrix}$
+                };
+
+                \draw[-{Latex}, line width=1pt] (model) -- (x);
+                \node[above=22mm of model.center] {Modell};
+                \node[above=20.8mm of x.center] {Beobachtung};
+            \end{tikzpicture}%
+            \vspace*{15mm}
+        \end{subfigure}%
+        \vspace*{-12.6mm}%
+        \begin{subfigure}{0.5\textwidth}
+            \centering
+
+            \begin{itemize}
+                \item Statistik
+            \end{itemize}
+            \begin{tikzpicture}
+                \node[
+                    rectangle,
+                    minimum width=7.5cm, minimum height=4.5cm,
+                    line width=1pt,
+                    draw=kit-orange, fill=kit-orange!20,
+                ] (real) {};
+
+                \node[right=of real] (x) {
+                    $\bm{x} =
+                    \begin{pmatrix}
+                        x_1 \\
+                        \vdots \\
+                        x_N
+                    \end{pmatrix}$
+                };
+
+                \draw[-{Latex}, line width=1pt] (real) -- (x);
+                \node[above=23mm of real.center] {``Echte Welt''};
+                \node[above=21.8mm of x.center] {Beobachtung};
+
+                \node[
+                    rectangle,
+                    minimum width=6.5cm, minimum height=3.5cm,
+                    line width=1pt,
+                    draw=kit-blue, fill=kit-blue!20,
+                    densely dashed,
+                ] (model) at (real) {
+                    $\bm{X} =
+                    \begin{pmatrix}
+                        X_1 \\
+                        \vdots \\
+                        X_N
+                    \end{pmatrix}\sim P_{\bm{X}}$
+                };
+
+                \draw[
+                    line width=1pt, densely dashed,
+                ] (x.south)
+                edge[-{Latex}, bend left]
+                node[below] {Modellierung}
+                (model.south);
+            \end{tikzpicture}
+            \vspace*{1mm}
+        \end{subfigure}
+    \end{figure}
+\end{frame}
+\else
 \begin{frame}
     \frametitle{Wahrscheinlichkeitstheorie und Statistik}
 
@@ -280,7 +397,81 @@
         }
     \end{figure}
 \end{frame}
+\fi
 
+\ifdefined\ishandout
+\begin{frame}
+    \frametitle{Punktschätzer}
+
+    \vspace*{-10mm}
+
+    \begin{itemize}
+        \item Beispiel: Temperaturschätzung
+            \vspace*{-5mm}
+            \begin{figure}[H]
+                \centering
+
+                \begin{tikzpicture}
+                    \node[
+                        rectangle,
+                        densely dashed,
+                        draw,
+                        inner sep=5mm,
+                    ] (x) {
+                        $
+                        \bm{x} =
+                        \begin{pmatrix}
+                            26{,}2 \\
+                            27{,}8 \\
+                            25{,}7 \\
+                            \vdots
+                        \end{pmatrix}
+                        $
+                    };
+
+                    \node[
+                        rectangle,
+                        right=of x,
+                        minimum width=5cm, minimum height=2cm,
+                        draw=kit-green, fill=kit-green!20,
+                        line width=1pt,
+                        align=center,
+                        inner sep=3mm
+                    ] (est) {Schätzer\\[5mm] $T_N(\bm{x}) =
+                        \displaystyle\frac{1}{N}
+                    \nsum_{i=0}^{N} x_i$};
+
+                    \node[
+                        above=of est,
+                        rectangle,
+                        densely dashed,
+                        draw,
+                        inner sep=5mm,
+                    ] (model) {
+                        $X_i \sim \mathcal{N}(\mu = \vartheta,
+                        \sigma^2 = 1)$
+                    };
+
+                    \node[right=of est] (theta) {$\hat{\vartheta}
+                    = 26{,}0$};
+
+                    \node[below] at (x.south) {Beobachtung};
+                    \node[above] at (model.north) {Parametrisiertes Modell};
+
+                    \draw[-{Latex}, line width=1pt] (x) -- (est);
+                    \draw[-{Latex}, line width=1pt] (model) -- (est);
+                    \draw[-{Latex}, line width=1pt] (model) -- (est);
+                    \draw[-{Latex}, line width=1pt] (est) -- (theta);
+                \end{tikzpicture}
+            \end{figure}
+        \item Punktschätzer: Rechenvorschrift zur Berechnung von
+            Parametern aus Beobachtungen \\
+            $\rightarrow$ Schätzer hängen von den Realisierungen ab
+            und sind damit selbst auch zufällig \\
+            $\rightarrow$ Schätzer haben einen Erwartungswert und eine Varianz
+    \end{itemize}
+\end{frame}
+\else
 \begin{frame}
     \frametitle{Punktschätzer}
 
@@ -463,6 +654,7 @@
             $\rightarrow$ Schätzer haben einen Erwartungswert und eine Varianz
     \end{itemize}
 \end{frame}
+\fi
 
 \begin{frame}
     \frametitle{Likelihood und Log-Likelihood (Diskret)}
@@ -601,7 +793,7 @@
                     Cramér-Rao Ungleichung \\
                     \vspace*{-6mm}
                     \begin{gather*}
-                        V(\hat{\vartheta}) \le \frac{1}{J(\vartheta)}
+                        V(\hat{\vartheta}) \ge \frac{1}{J(\vartheta)}
                     \end{gather*}
                     \vspace*{-10mm}
                 \end{lightgrayhighlightbox}
@@ -820,7 +1012,7 @@
             \end{minipage}
             \begin{minipage}{0.16\textwidth}
                 \begin{gather*}
-                    E\left( \lvert \hat{\lambda}_\text{ML} - \lambda
+                    P\left( \lvert \hat{\lambda}_\text{ML} - \lambda
                         \rvert \ge \varepsilon
                     \right)
                 \end{gather*}
@@ -828,7 +1020,7 @@
             \pause %
             \begin{minipage}{0.22\textwidth}
                 \begin{gather*}
-                    = E\left( \lvert \hat{\lambda}_\text{ML} -
+                    = P\left( \lvert \hat{\lambda}_\text{ML} -
                         E\left(\hat{\lambda}_\text{ML}\right) \rvert
                         \ge \varepsilon
                     \right)
@@ -846,8 +1038,8 @@
             \end{gather*}
             \pause
             \begin{gather*}
-                E\left( \lvert \hat{\lambda}_\text{ML} - \lambda
-                    \rvert > \varepsilon
+                P\left( \lvert \hat{\lambda}_\text{ML} - \lambda
+                    \rvert \ge \varepsilon
                 \right) \le \frac{\lambda}{N \varepsilon^2}
                 \overset{N\rightarrow
                 \infty}{\relbar\joinrel\relbar\joinrel\relbar\joinrel\rightarrow}