tut4: Started adding theory for exercise 1

src/2025-12-19/presentation.tex

@@ -81,7 +81,90 @@
 \subsection{Theorie Wiederholung}
 
 \begin{frame}
-    \frametitle{sasdf}
+    \frametitle{Stetige Zufallsvariablen}
+
+    \vspace*{-10mm}
+
+    \begin{lightgrayhighlightbox}
+        Erinnerung: Diskrete Zufallsvariablen
+        \begin{align*}
+            \text{\normalfont Verteilung: }& P_X(x) = P(X = x) \\
+            \text{\normalfont Verteilungsfunktion: }& F_X(x) = P(X \le x) =
+            \sum_{n: x_n \le y} P_X(x)
+        \end{align*}
+        \vspace{-10mm}
+    \end{lightgrayhighlightbox}
+
+    \begin{columns}[t]
+        \pause\column{\kitthreecolumns}
+        \centering
+        \begin{itemize}
+            \item Verteilungsfunktion $F_X(x)$ einer stetiger ZV
+                \begin{gather*}
+                    F_X(x) = P(X \le x) \\[5mm]
+                    \text{Eigenschaften:} \\[3mm]
+                    \lim_{x\rightarrow -\infty} F_X(x) = 0 \\
+                    \lim_{x\rightarrow +\infty} F_X(x) = 1 \\
+                    x_1 \le x_2 \Rightarrow F_X(x_1) \le F_X(x_2)\\
+                    \lim_{h\rightarrow 0^+} F_X(x + h) = F_X(x)
+                \end{gather*}
+        \end{itemize}
+        \pause\column{\kitthreecolumns}
+        \centering
+        \begin{itemize}
+            \item Wahrscheinlichkeitsdichte $f_X(x)$ einer stetiger ZV
+                \begin{gather*}
+                    F_X(x) = \int_{-\infty}^{x} f_X(u) du \\[5mm]
+                    \text{Eigenschaften:} \\[3mm]
+                    f_X(x) \ge 0 \\
+                    \int_{-\infty}^{\infty} f_X(x) dx = 1
+                \end{gather*}
+        \end{itemize}
+    \end{columns}
+\end{frame}
+
+% TODO: Write this
+\begin{frame}
+    \frametitle{TODO}
+
+\end{frame}
+
+\begin{frame}
+    \frametitle{Zusammenfassung}
+
+    \begin{columns}[c]
+        \column{\kitthreecolumns}
+        \centering
+        \begin{greenblock}{Verteilungsfunktion (kontinuierlich)}
+            \vspace*{-6mm}
+            \begin{gather*}
+                F_X(x) = P(X \le x)\\[8mm]
+                \lim_{x\rightarrow -\infty} F_X(x) = 0 \\
+                \lim_{x\rightarrow +\infty} F_X(x) = 1 \\
+                x_1 \le x_2 \Rightarrow F_X(x_1) \le F_X(x_2)\\
+                \lim_{h\rightarrow 0^+} F_X(x + h) = F_X(x)
+            \end{gather*}
+        \end{greenblock}
+        \column{\kitthreecolumns}
+        \centering
+        \begin{greenblock}{Wahrscheinlichkeitsdichte \phantom{()}}
+            \vspace*{-6mm}
+            \begin{gather*}
+                F_X(x) = \int_{-\infty}^{x} f_X(u) du \\[5mm]
+                f_X(x) \ge 0 \\
+                \int_{-\infty}^{\infty} f_X(x) dx = 1
+            \end{gather*}
+        \end{greenblock}
+        %TODO: Rename this
+        \begin{greenblock}{TODO}
+            \vspace*{-6mm}
+            \begin{gather*}
+                P(a < X \le b) = F_X(b) - F_X(a) \\[2mm]
+                E(X) = \int_{-\infty}^{x} u f_X(u) du
+            \end{gather*}
+        \end{greenblock}
+
+    \end{columns}
 \end{frame}
 
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