diff --git a/src/2026-02-13/presentation.tex b/src/2026-02-13/presentation.tex index 372b211..834ac6d 100644 --- a/src/2026-02-13/presentation.tex +++ b/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}