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Add summary slide for exercise 1

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Andreas Tsouchlos 2026-01-15 03:38:23 +01:00
parent 876bbad136
commit 8eb3a6378f
1 changed files with 80 additions and 45 deletions
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125
src/2026-01-16/presentation.tex
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@ -96,7 +96,42 @@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Theorie Wiederholung}
% TODO:
\begin{frame}
\frametitle{Zusammenfassung}
\begin{columns}[t]
\column{\kitthreecolumns}
\begin{greenblock}{Poisson Verteilung}
\vspace*{-6mm}
\begin{gather*}
X \sim \text{Poisson}(\lambda) \\
P_X(k) = \frac{\lambda^k \cdot e^{-\lambda}}{k!}
\end{gather*}
\end{greenblock}
\begin{greenblock}{Binomialentwicklung}
\vspace*{-6mm}
\begin{gather*}
\nsum_{k=0}^{n} \binom{n}{k}a^k b^{n-k} = (a+b)^n, \hspace{15mm}
\binom{n}{k} = \frac{n!}{(n-k!)k!}
\end{gather*}
\end{greenblock}
\column{\kitthreecolumns}
\begin{greenblock}{Faltungssatz}
\vspace*{-6mm}
\begin{gather*}
Z = X + Y \\
P_Z(n) = \nsum_{k=0}^{n} P_X(k)P_Y(n-k)
\end{gather*}
\end{greenblock}
\begin{greenblock}{Charakteristische Funktion einer Summe von ZVs}
\vspace*{-6mm}
\begin{gather*}
Z = X + Y \\
\phi_Z(s) = \phi_X(s) \cdot \phi_Y(s)
\end{gather*}
\end{greenblock}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Aufgabe}
@ -390,50 +425,50 @@
\begin{figure}[H]
\centering
\begin{tikzpicture}
\begin{axis}[
view={20}{30},
xlabel=$x$, ylabel=$y$, zlabel={$f_{X,Y}(x,y)$},
xmin=0, xmax=1, ymin=0, ymax=1, zmin=0, zmax=2,
xtick={0,0.5,1},ytick={0,0.5,1},ztick={0,1,2},
point meta min=0, point meta max=2,
declare function={cutoff(\x) = 0.3/\x;},
legend,
]
\addplot3[
surf, shader=interp,
samples=40,
domain=0:1, y domain=0:1
] (
x,
{y * min(1, cutoff(x))},
{x + (y * min(1, cutoff(x)))}
);
\addlegendentry{$x\cdot y \le z$}
\addplot3[
surf, shader=interp,
samples=40,
domain=0.3:1, y domain=0:1,
fill=gray,
draw=none,
point meta=1.1,
colormap name=cividis,
] (
x,
{cutoff(x) + y*(1 - cutoff(x))},
{x + (cutoff(x) + y*(1 - cutoff(x)))}
);
\addplot3[
mesh,
samples=15,
domain=0:1, y domain=0:1,
draw=black,
opacity=0.3
] {x + y};
\end{axis}
\end{tikzpicture}
% \begin{tikzpicture}
% \begin{axis}[
% view={20}{30},
% xlabel=$x$, ylabel=$y$, zlabel={$f_{X,Y}(x,y)$},
% xmin=0, xmax=1, ymin=0, ymax=1, zmin=0, zmax=2,
% xtick={0,0.5,1},ytick={0,0.5,1},ztick={0,1,2},
% point meta min=0, point meta max=2,
% declare function={cutoff(\x) = 0.3/\x;},
% legend,
% ]
% \addplot3[
% surf, shader=interp,
% samples=40,
% domain=0:1, y domain=0:1
% ] (
% x,
% {y * min(1, cutoff(x))},
% {x + (y * min(1, cutoff(x)))}
% );
% \addlegendentry{$x\cdot y \le z$}
%
% \addplot3[
% surf, shader=interp,
% samples=40,
% domain=0.3:1, y domain=0:1,
% fill=gray,
% draw=none,
% point meta=1.1,
% colormap name=cividis,
% ] (
% x,
% {cutoff(x) + y*(1 - cutoff(x))},
% {x + (cutoff(x) + y*(1 - cutoff(x)))}
% );
%
% \addplot3[
% mesh,
% samples=15,
% domain=0:1, y domain=0:1,
% draw=black,
% opacity=0.3
% ] {x + y};
% \end{axis}
% \end{tikzpicture}
\end{figure}
\end{minipage}%
\begin{minipage}{0.58\textwidth}