diff --git a/src/2026-01-16/presentation.tex b/src/2026-01-16/presentation.tex index b6019e5..05639f1 100644 --- a/src/2026-01-16/presentation.tex +++ b/src/2026-01-16/presentation.tex @@ -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}