From 8dad61d27a6998b82d51e8f94d6fedb87f48302c Mon Sep 17 00:00:00 2001
From: Andreas Tsouchlos <an.tsouchlos@gmail.com>
Date: Wed, 14 Jan 2026 01:30:12 +0100
Subject: [PATCH] Finish solution for exercise 2a

---
 src/2026-01-16/presentation.tex | 61 ++++++++++++++++++++-------------
 1 file changed, 37 insertions(+), 24 deletions(-)

diff --git a/src/2026-01-16/presentation.tex b/src/2026-01-16/presentation.tex
index 61f9b6b..5e05529 100644
--- a/src/2026-01-16/presentation.tex
+++ b/src/2026-01-16/presentation.tex
@@ -249,7 +249,16 @@
                     X = h_1(U,V) = U \\
                     Y = h_2(U,V) = \frac{V}{U}
                 \end{array}
-                \right. \\[4mm]
+                \hspace{20mm}
+                \left(\begin{array}{l}
+                    0 < x \le 1 \Rightarrow 0 < u \le 1 \\
+                    0 < y \le 1 \Rightarrow 0 < v \le u \le 1
+                \end{array}
+                \right)
+                \right.
+            \end{gather*}
+            \vspace*{5mm}
+            \pause\begin{gather}
                 \mathcal{J} = \begin{pmatrix}
                     \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v}  \\[2mm]
                     \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}
@@ -258,32 +267,37 @@
                      1 & 0 \\
                      - \frac{v}{u^2} & \frac{1}{u}
                 \end{pmatrix}
-            \end{gather*}
-            \vspace{3mm}
+            \end{gather}
             \begin{align*}
-                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)
-                             = \left\{ \begin{array}{ll}
-                                    \frac{1}{u} \cdot \left(u + \frac{v}{u}\right) &,\hspace{2mm} 0 < v \le u \le 1 \\
-                                    0 &,\hspace{2mm} \text{sonst}
-                                \end{array} \right.
-                             = \left\{ \begin{array}{ll}
-                                    1 + \frac{v}{u^2} &,\hspace{2mm} 0 < v \le u \le 1 \\
-                                    0 &,\hspace{2mm} \text{sonst}
-                                \end{array} \right.
+                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)
+                    = \frac{1}{u} \cdot \left(u + \frac{v}{u}\right)
+                = 1 + \frac{v}{u^2}, && \hspace*{-20mm} 0 < v \le u \le 1 \\[3mm]
             \end{align*}
-            \vspace{3mm}
-            \begin{gather*}
-                \text{Für } 0 < v \le 1: \hspace{5mm} f_V(v)
-                    = \int_{-\infty}^{\infty} f_{U,V}(u,v) du
-                    = \int_{v}^{1} 1 + \frac{v}{u^2} dx
+            \vspace*{-22mm}
+            \pause\begin{align*}
+                f_V(v) &= \int_{-\infty}^{\infty} f_{U,V}(u,v) du
+                    = \int_{v}^{1} 1 + \frac{v}{u^2} du
                     = \left[ u - \frac{v}{u} \right]_v^1
-                    = 2(1-v) \\[3mm]
-                    f_Z(z) = \left\{\begin{array}{ll}
-                        2(1-z) \hspace{3mm}&,\hspace{3mm} 0 < z \le 1 \\
-                        0 \hspace{3mm}&,\hspace{3mm} \text{sonst}\\
-                    \end{array}\right.
+                    = 2(1-v), && \hspace*{-20mm} 0 < v \le 1
+            \end{align*}
+            \vspace{5mm}
+            \pause\begin{gather*}
+                f_Z(z) = \left\{\begin{array}{ll}
+                    2(1-z) \hspace{3mm}&,\hspace{3mm} 0 < z \le 1 \\
+                    0 \hspace{3mm}&,\hspace{3mm} \text{sonst}\\
+                \end{array}\right.
             \end{gather*}
-        \pause \item Verwenden Sie einen alternativen Ansatz zur Berechnung der
+    \end{enumerate}
+    % tex-fmt: on
+\end{frame}
+
+\begin{frame}
+    \frametitle{Aufgabe 2: Transformationssatz für 2D-Dichten}
+
+    \begin{enumerate}[a{)}]
+        \setcounter{enumi}{1}
+        \item Verwenden Sie einen alternativen Ansatz zur Berechnung der
             Dichte. Hinweis: Beginnen Sie mit $P (Z \le z) = \ldots$
             \pause\begin{align*}
             \end{align*}
@@ -291,7 +305,6 @@
             \pause\begin{align*}
             \end{align*}
     \end{enumerate}
-    % tex-fmt: on
 \end{frame}
 
 \end{document}