Finish theory for part 1

src/2025-12-19/presentation.tex

@@ -103,14 +103,6 @@
                 \begin{gather*}
                     F_X(x) = P(X \le x)
                 \end{gather*}
-                \pause\vspace{-10mm}
-                \begin{gather*}
-                    \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
@@ -119,50 +111,58 @@
                 \begin{gather*}
                     F_X(x) = \int_{-\infty}^{x} f_X(u) du
                 \end{gather*}
-                \pause\vspace{-10mm}
+        \end{itemize}
+    \end{columns}
+    \begin{columns}[t]
+        \pause \column{\kitthreecolumns}
+        \centering
+        \begin{gather*}
+            \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)\\
+            F_X(x+) = \lim_{h\rightarrow 0^+} F_X (x+h) = F_X(x)
+        \end{gather*}
+        \pause \column{\kitthreecolumns}
+        \centering
         \begin{gather*}
             \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
-% TODO: P(a < X < b) = F_X(b) - F_X(a)
-% TODO: E(X_cont) = int(...) vs E(X_disc) = sum(...)
 \begin{frame}
     \frametitle{Stetige Zufallsvariablen II}
 
-    \begin{columns}
+    \begin{minipage}{0.6\textwidth}
-        \column{\kitfourcolumns}
-        \centering
         \begin{itemize}
-            \item Kenngrößen
+            \item Wichtige Kenngrößen
                 \begin{align*}
                     \begin{array}{rlr}
                         \text{Erwartungswert: } \hspace{5mm} & E(X) =
                         \displaystyle\int_{-\infty}^{\infty} x f_X(x) dx
-                        & \hspace{5mm} \big( = \mu  \big) \\
+                        & \hspace{5mm} \big( = \mu  \big) \\[3mm]
                         \text{Varianz: } \hspace{5mm} & V(X) = E\mleft(
-                        \mleft( X - E(X) \mright)^2 \mright) \\
+                        \mleft( X - E(X) \mright)^2 \mright) \\[3mm]
                         \text{Standardabweichung: } \hspace{5mm} &
                         \sqrt{V(X)} & \hspace{5mm} \big( = \sigma \big)
                     \end{array}
                 \end{align*}
         \end{itemize}
-        \column{\kittwocolumns}
+    \end{minipage}
-        \centering
+    \begin{minipage}{0.38\textwidth}
         \begin{lightgrayhighlightbox}
             Erinnerung
             \begin{align*}
-                \text{\normalfont Erwartungswert: }& E(X) = \sum_{n=1}^{\infty} x_n P_X(x) \\ 
+                \text{\normalfont Erwartungswert: }& E(X) =
-                \text{\normalfont Varianz: }& V(X) = E\mleft( \mleft( X - E(X) \mright)^2 \mright)
+                \sum_{n=1}^{\infty} x_n P_X(x) \\
+                \text{\normalfont Varianz: }& V(X) = E\mleft( \mleft(
+                X - E(X) \mright)^2 \mright)
             \end{align*}
         \end{lightgrayhighlightbox}
-    \end{columns}
+    \end{minipage}
-
 \end{frame}
 
 \begin{frame}
@@ -174,11 +174,12 @@
         \begin{greenblock}{Verteilungsfunktion (kontinuierlich)}
             \vspace*{-6mm}
             \begin{gather*}
-                F_X(x) = P(X \le x)\\[8mm]
+                F_X(x) = P(X \le x)\\[4mm]
+                P(a < X \le b) = F_X(b) - F_X(a) \\[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)
+                F_X(x+) = \lim_{h\rightarrow 0^+} F_X (x+h) = F_X(x)
             \end{gather*}
         \end{greenblock}
         \column{\kitthreecolumns}
@@ -191,15 +192,6 @@
                 \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}
 
@@ -339,7 +331,6 @@
                 \begin{gather*}
                     x_1 \le x_2 \Rightarrow F_X(x_1) \le F_X(x_2) \\
                     F_X(x+) = \lim_{h\rightarrow 0^+} F_X (x+h) = F_X(x)
-                    \hspace{5mm}\forall x\in \mathbb{R}
                 \end{gather*}
                 \column{\kitonecolumn}
             \end{columns}
@@ -386,7 +377,7 @@
                 = P(1 < X \le 2) = F_X(2) - F_X(1) = e^{-a} - e^{-4a}
             \end{gather*}
     \end{enumerate}
-    % tex-fmt: off
+    % tex-fmt: on
 \end{frame}
 
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