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Finish theory for exercise 2

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Andreas Tsouchlos
2026-02-13 02:02:51 +01:00
parent 948523c1b5
commit 2e25934348
1 changed files with 166 additions and 3 deletions
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src/2026-02-13/presentation.tex
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@@ -39,6 +39,7 @@
\pgfplotsset{compat=newest}
\usepgfplotslibrary{fillbetween}
\usepgfplotslibrary{groupplots}
\usepgfplotslibrary{statistics}
\usepackage{enumerate}
\usepackage{listings}
@@ -568,7 +569,7 @@
\pause
\item Konsistenz
\begin{gather*}
\lim_{N\rightarrow \infty} P_\vartheta \big( \lvert
\lim_{N\rightarrow \infty} P \big( \lvert
\hat{\vartheta} - \vartheta \rvert \ge \varepsilon \big) = 0
\end{gather*}
@@ -980,6 +981,7 @@
+ x_{\left( Np + 1 \right)} \right), & Np \in \mathbb{N}
\end{cases}
\end{align*}
\pause
\item Quartilsabstand
\begin{align*}
x_{3/4} - x_{1/4}
@@ -990,7 +992,169 @@
\begin{frame}
\frametitle{Boxplots}
% TODO: Create slide
\vspace*{-10mm}
\begin{figure}[H]
\centering
\begin{tikzpicture}
\begin{axis}[
width=24cm,
height=6cm,
clip=false,
xticklabel=\empty,
yticklabel=\empty,
]
\addplot+ [
mark=*,
kit-red,
boxplot prepared={
lower whisker=5,
lower quartile=7,
median=8.5,
upper quartile=9.5,
upper whisker=10,
},
boxplot/every median/.style={draw=kit-blue,line width=2pt},
boxplot/every whisker/.style={draw=kit-green,line
width=1pt},
boxplot/every box/.style={black,line width=1pt},
] table [row sep=\\,y index=0] {
data\\ 1\\ 3\\
};
\node at (7.5,0) (median)
{\textcolor{kit-blue}{Median: $x_{1/2}$}};
\node[below right=0cm and 0cm of median,align=center]
(uw) {
\textcolor{kit-green}{Größte normale Beobachtung:}\\
$\textcolor{kit-green}{x_{3/4} + \frac{3}{2}
\left( x_{3/4} - x_{1/4} \right)}$
};
\node[below left=0cm and 0cm of median,align=center] (lw)
{
\textcolor{kit-green}{Kleinste normale Beobachtung:}\\
$\textcolor{kit-green}{x_{1/4} - \frac{3}{2} \left(
x_{3/4} - x_{1/4} \right)}$
};
\node at (9.78, 2) (uq)
{Oberes Quartil: $x_{3/4}$};
\node[left=of uq] (lq)
{Unteres Quartil: $x_{1/4}$};
\node[above left=0cm and 0cm of lw] (out)
{\textcolor{kit-red}{Ausreißer}};
\draw[kit-blue, line width=1pt] (axis cs: 8.4,0.65) -- (median);
\draw[kit-green, line width=1pt] (axis cs: 5,0.65) -- (lw);
\draw[kit-green, line width=1pt] (axis cs: 10,0.65) -- (uw);
\draw[kit-red, line width=1pt] (axis cs: 1.08,0.9) -- (out);
\draw[kit-red, line width=1pt] (axis cs: 2.9,0.9) -- (out);
\draw[line width=1pt] (axis cs: 7,1.42) -- (lq);
\draw[line width=1pt] (axis cs: 9.5,1.42) -- (uq);
\end{axis}
\end{tikzpicture}
\end{figure}
\end{frame}
\begin{frame}
\frametitle{Zusammenfassung}
\vspace*{-10mm}
\begin{columns}
\column{\kitfourcolumns}
\begin{greenblock}{Empirische Kenngrößen}
\vspace*{-8mm}
\begin{align*}
\text{Empirischer Erwartungswert: } & \overline{x} = \frac{1}{N}
\nsum_{i=1}^{N} x_i \\[3mm]
\text{Empirische Varianz: } & s^2 = \frac{1}{N-1}
\nsum_{i=1}^{N} \left( x_i - \overline{x} \right)^2 \\[3mm]
p\text{-Quantil: }
& 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} \\[3mm]
\text{Median: } & x_{1/2} \\[3mm]
\text{Quartilsabstand: } & x_{3/4} - x_{1/4}
\end{align*}
\vspace*{-4mm}
\end{greenblock}
\column{\kittwocolumns}
\begin{greenblock}{Boxplot}
\begin{figure}[H]
\centering
\begin{tikzpicture}
\begin{axis}[
width=3cm,
height=12cm,
boxplot/draw direction=y,
clip=false,
xtick=\empty,
ytick=\empty,
axis lines=none,
]
\addplot+ [
mark=*,
kit-red,
boxplot prepared={
lower whisker=5,
lower quartile=7,
median=8.5,
upper quartile=9.5,
upper whisker=10,
},
boxplot/every
median/.style={draw=kit-blue,line width=2pt},
boxplot/every
whisker/.style={draw=kit-green,line width=1pt},
boxplot/every box/.style={black,line width=1pt},
] table [row sep=\\,y index=0] {
data\\ 1\\ 3\\
};
\node[right] (median) at (2.5,8.5)
{$\textcolor{kit-blue}{x_{1/2}}$};
\node[right] (lq) at (2.5,7) {$x_{1/4}$};
\node[right] (uq) at (2.5,9.5) {$x_{3/4}$};
\node[right] (lw) at (2.5,5)
{
$\textcolor{kit-green}{
x_{1/4} - \frac{3}{2} \left( x_{3/4} -
x_{1/4} \right)
}$
};
\node[right] (uw) at (2.5,10.6)
{
$\textcolor{kit-green}{
x_{1/4} + \frac{3}{2} \left( x_{3/4} -
x_{1/4} \right)
}$
};
\node[right] (out) at (2.5,2)
{
\textcolor{kit-red}{Ausreißer}
};
\draw[kit-blue,line width=1pt] (1.6,8.5) -- (median);
\draw[line width=1pt] (1.6,9.5) -- (uq);
\draw[line width=1pt] (1.6,7) -- (lq);
\draw[kit-green,line width=1pt] (1.6,5) -- (lw);
\draw[kit-green,line width=1pt] (1.6,10) -- (uw);
\draw[kit-red,line width=1pt] (1.3,1) -- (out);
\draw[kit-red,line width=1pt] (1.3,3) -- (out);
\end{axis}
\end{tikzpicture}
\end{figure}
\end{greenblock}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -1125,7 +1289,6 @@
\end{align*}
\end{frame}
% TODO: p-Quantil einer Stichprobe [p. 113]
\begin{frame}
\frametitle{Aufgabe 2: Deskriptive Statistik}