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Add slide explaining marginals and transformations

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Andreas Tsouchlos 2026-01-16 01:21:48 +01:00
parent 33ff39f974
commit 54407061a0
1 changed files with 109 additions and 11 deletions
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src/2026-01-16/presentation.tex
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@ -35,6 +35,7 @@
\usepackage{pgfplots}
\pgfplotsset{compat=newest}
\usepgfplotslibrary{fillbetween}
\usepgfplotslibrary{groupplots}
\usepackage{enumerate}
\usepackage{listings}
@ -107,7 +108,7 @@
\begin{itemize}
\item Unabhängigkeit hat nichts mit den Einzelverteilungen zu
tun, sie ist "eine Ebene höher"
tun, sie ist ``eine Ebene höher''
\end{itemize}
\end{frame}
@ -300,11 +301,108 @@
\begin{frame}
\frametitle{Mehrdimensionale Zufallsvariablen}
\begin{itemize}
\item Randdichte
\item Transformationssatz (betonen, dass h1, h2 eineindeutig
sein müssen; Bild von Folie 85)
\end{itemize}
\vspace*{-20mm}
\begin{columns}[t]
\column{\kitfourcolumns}
\begin{itemize}
\item Randdichte
\begin{align*}
f_X(x) = \int_{-\infty}^{\infty} f_{X,Y}(x,y) dy
\end{align*}
\end{itemize}
\column{\kittwocolumns}
\begin{figure}[H]
\centering
\begin{tikzpicture}[
/pgfplots/scale only axis,
/pgfplots/width=5cm,
/pgfplots/height=5cm
]
\begin{axis}[
name=main axis,
view={0}{90},
ticks=none,
xlabel={$x$},ylabel={$y$},
]
\addplot3[
surf, shader=interp,
samples=40,
domain=-3:3, y domain=-3:3
]
{1/(2*pi*sqrt(0.5)) * exp(-1/(2*(1 -
sqrt(0.5))) * (x^2 -2*sqrt(0.5)*x*y + y^2) )};
\end{axis}
\node[below] at
($(main axis.south west) + (-.5, -.5)$) {$f_{X,Y}(x,y)$};
\begin{axis}[
anchor=south west,
at=(main axis.north west),
height=2cm,
ticks=none,
ylabel={$f_X(x)$},
samples=50,
domain=-3:3,
xmin=-3,xmax=3,
]
\addplot[line width=1pt] {1/sqrt(2*pi) *
exp(-x^2/2)};
\end{axis}
\begin{axis}[
anchor=north west,
at=(main axis.north east),
width=2cm,
ticks=none,
xlabel={$f_Y(y)$},
samples=50,
domain=-3:3,
ymin=-3,ymax=3,
]
\addplot[line width=1pt] ( {1/sqrt(2*pi)
* exp(-x^2/2)}, {x} );
\end{axis}
\end{tikzpicture}
\end{figure}
\end{columns}
\pause
\vspace*{-45mm}
\begin{columns}
\column{\kitfourcolumns}
\begin{itemize}
\item Umrechnung von Dichten mit dem Transformationssatz
\begin{gather*}
X = h_1(U,V), \hspace{5mm} Y = h_2(U,V) \\[5mm]
\mathcal{J} =
\begin{pmatrix}
\frac{\displaystyle \partial}{\displaystyle
\partial u}x & \frac{\displaystyle
\partial}{\displaystyle \partial v}x \\[3mm]
\frac{\displaystyle \partial}{\displaystyle
\partial u}y & \frac{\displaystyle
\partial}{\displaystyle \partial v}y
\end{pmatrix}
=
\begin{pmatrix}
\frac{\displaystyle \partial}{\displaystyle
\partial u}h_1(u,v) & \frac{\displaystyle
\partial}{\displaystyle \partial v}h_1(u,v) \\[3mm]
\frac{\displaystyle \partial}{\displaystyle
\partial u}h_2(u,v) & \frac{\displaystyle
\partial}{\displaystyle \partial v}h_2(u,v)
\end{pmatrix} \\[5mm]
f_{U,V}(u,v) = \lvert
\text{det}(\mathcal{J}) \rvert
\cdot f_{X,Y} \big(h_1(u,v),h_2(u,v)\big)
\end{gather*}
\end{itemize}
\column{\kittwocolumns}
\end{columns}
\end{frame}
\begin{frame}
@ -333,15 +431,15 @@
\end{gather*}
\end{greenblock}
\column{\kitfourcolumns}
\begin{greenblock}{Transformationssatz}
\begin{greenblock}{Umrechnung von Dichten mit dem Transformationssatz}
\vspace*{-6mm}
\begin{gather*}
X = h_1(U,V), \hspace{5mm} Y = h_2(U,V) \\[2mm]
X = h_1(U,V), \hspace{5mm} Y = h_2(U,V) \\[5mm]
\mathcal{J} =
\begin{pmatrix}
\frac{\displaystyle \partial}{\displaystyle
\partial u}x & \frac{\displaystyle
\partial}{\displaystyle \partial v}x \\[2mm]
\partial}{\displaystyle \partial v}x \\[3mm]
\frac{\displaystyle \partial}{\displaystyle
\partial u}y & \frac{\displaystyle
\partial}{\displaystyle \partial v}y
@ -350,11 +448,11 @@
\begin{pmatrix}
\frac{\displaystyle \partial}{\displaystyle
\partial u}h_1(u,v) & \frac{\displaystyle
\partial}{\displaystyle \partial v}h_1(u,v) \\[2mm]
\partial}{\displaystyle \partial v}h_1(u,v) \\[3mm]
\frac{\displaystyle \partial}{\displaystyle
\partial u}h_2(u,v) & \frac{\displaystyle
\partial}{\displaystyle \partial v}h_2(u,v)
\end{pmatrix} \\[3mm]
\end{pmatrix} \\[5mm]
f_{U,V}(u,v) = \lvert
\text{det}(\mathcal{J}) \rvert
\cdot f_{X,Y} \big(h_1(u,v),h_2(u,v)\big)