an.tsouchlos/wt-tut-presentations
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases 21 Wiki Activity

Add solution for 2b

Browse Source
This commit is contained in:
Andreas Tsouchlos 2026-01-15 03:26:33 +01:00
parent 587d894e5e
commit 876bbad136
1 changed files with 111 additions and 15 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

126
src/2026-01-16/presentation.tex
View File

@ -239,7 +239,8 @@
\partial u}y & \frac{\displaystyle
\partial}{\displaystyle \partial v}y
\end{pmatrix}
= \begin{pmatrix}
=
\begin{pmatrix}
\frac{\displaystyle \partial}{\displaystyle
\partial u}h_1(u,v) & \frac{\displaystyle
\partial}{\displaystyle \partial v}h_1(u,v) \\[2mm]
@ -357,20 +358,115 @@
\begin{frame}
\frametitle{Aufgabe 2: Transformationssatz für 2D-Dichten}
% tex-fmt: off
\begin{enumerate}[a{)}]
\setcounter{enumi}{1}
\item Verwenden Sie einen alternativen Ansatz zur Berechnung der
Dichte.\\
\textit{Hinweis}: Beginnen Sie mit $P (Z \le z) = \ldots$
\pause\begin{align*}
P(Z \le z) = P(XZ \le z) &= \int_{-\infty}^{\infty}
\int_{-\infty}^{z/x} f_{X,Y}(x,y) dy dx \\[1mm]
&= \int_{-\infty}^{\infty} \int_{-\infty}^{z}
f_{X,Y}\left(x, \frac{u}{x}\right) \frac{1}{x} \; du dx
\end{align*}
\end{enumerate}
% tex-fmt: on
\begin{minipage}[c]{0.64\textwidth}
% tex-fmt: off
\begin{enumerate}[a{)}]
\setcounter{enumi}{1}
\item Verwenden Sie einen alternativen Ansatz zur Berechnung der
Dichte.\\
\textit{Hinweis}: Beginnen Sie mit $P (Z \le z) = \ldots$
\end{enumerate}
% tex-fmt: on
\end{minipage}%
\begin{minipage}[c]{0.35\textwidth}
\begin{lightgrayhighlightbox}
\vspace*{-8mm}
% tex-fmt: off
\begin{gather*}
\text{Bekannt: } \hspace{10mm} f_{X,Y}(x,y) = x + y
\end{gather*}
% tex-fmt: on
\vspace*{-12mm}
\end{lightgrayhighlightbox}
\end{minipage}
\pause
\begin{align*}
P(Z \le z) = \int_{-\infty}^{z} f_Z(t) dt
\end{align*}
\begin{minipage}{0.4\textwidth}
\pause
\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}
\end{figure}
\end{minipage}%
\begin{minipage}{0.58\textwidth}
\pause
\begin{align*}
P(Z \le z) &= P(XY \le z) = \int_{-\infty}^{\infty}
\int_{-\infty}^{z/x} f_{X,Y}(x,y) dy dx
\end{align*}
\vspace*{-10mm}
\pause
\begin{align*}
\overset{
\begin{subarray}{l}
u = xy \\
du = xdy
\end{subarray}}{=}
&\int_{-\infty}^{\infty} \int_{-\infty}^{z} f_{X,Y}(x,
\frac{u}{x})\frac{1}{x}\; du dx \\[2mm]
= &\int_{-\infty}^{z}
\underbrace{\int_{-\infty}^{\infty} f_{X,Y}(x,
\frac{u}{x})\frac{1}{x}\; dx}_{f_Z(u)}du \\
\end{align*}
\end{minipage}
\pause
\begin{gather*}
0 < y \le 1 \hspace{5mm} \Rightarrow\hspace{5mm} 0 <
\frac{u}{x} \le 1 \hspace{5mm}\Rightarrow\hspace{5mm} 0 <
u \le x \le 1 \\
f_Z(u) = \int_{-\infty}^{\infty} f_{X,Y}(x,
\frac{u}{x})\frac{1}{x}\; dx
= \int_{z}^{1} 1 + \frac{u}{x^2} dx = 2(1-u), \hspace{5mm} 0 < u \le 1
\end{gather*}
\end{frame}
\begin{frame}