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Add slide explaining prob. theory vs statistics

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Andreas Tsouchlos
2026-02-10 16:33:15 +01:00
parent aa159c8777
commit 40ff354ed9
1 changed files with 171 additions and 4 deletions
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175
src/2026-02-13/presentation.tex
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@@ -114,7 +114,172 @@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Theorie Wiederholung}
% TODO: Add slides
\begin{frame}
\frametitle{Wahrscheinlichkeitstheorie und Statistik}
\vspace*{-5mm}
\begin{itemize}
\item Einfache Stichprobe
\begin{gather*}
X_1, \ldots, X_N \hspace{2mm}\overbrace{\text{unabhäng 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}
\pause
\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}%
\only<2>{\hspace*{16cm}}%
\only<3->{\vspace*{-12.6mm}}%
\begin{subfigure}{0.5\textwidth}
\centering
\only<3>{
\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[below=25.5mm of real.center]
{\phantom{Modellierung}};
\end{tikzpicture}
}%
\only<4->{
\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}
\only<3->{
\vspace*{12.5mm}
}
\end{figure}
\end{frame}
\begin{frame}
\frametitle{Schätzer I}
\begin{itemize}
\item asdf
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Schätzer II}
\begin{itemize}
\item asdf
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Aufgabe}
@@ -139,13 +304,14 @@
[a{)}]
% tex-fmt: on
\item Bestimmen Sie die Log-Likelihoodfunktion für $N$
Messwerte und damit den ML-Schätzer für die Ankunftsrate $\lambda$.
Messwerte und damit den ML-Schätzer für die
Ankunftsrate $\lambda$.
\item Zeigen Sie, dass der Schätzer erwartungstreu ist.
\item Ist der ML-Schätzer konsistent?
\item Ist der ML-Schätzer effizient?
\end{enumerate}
\end{frame}
% TODO: Add slides
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -198,7 +364,8 @@
\end{gather*}
Durch fehlerhaftes Eintragen wurde für zwei weitere Studierende
die Studiendauer $0$ und $129$ vermerkt. Die erweiterte Stichprobe lautet:
die Studiendauer $0$ und $129$ vermerkt. Die erweiterte
Stichprobe lautet:
\vspace*{-5mm}
\begin{gather*}
z_1 =