Fix factorial symbol and change variable k to n

.latexmkrc

@@ -1,4 +1,3 @@
 $pdflatex="pdflatex -shell-escape -interaction=nonstopmode -synctex=1 %O %S";
 $out_dir = 'build';
 $pdf_mode = 1;
-

Makefile

@@ -1,19 +1,25 @@
 PRESENTATIONS := $(patsubst src/%/presentation.tex,build/presentation_%.pdf,$(wildcard src/*/presentation.tex))
 HANDOUTS := $(patsubst build/presentation_%.pdf,build/presentation_%_handout.pdf,$(PRESENTATIONS))
 
+RC_PDFLATEX := $(shell grep '$$pdflatex' .latexmkrc \
+	       | sed -e 's/.*"\(.*\)".*/\1/' -e 's/%S//' -e 's/%O//')
+
 .PHONY: all
 all: $(PRESENTATIONS) $(HANDOUTS)
 
 build/presentation_%.pdf: src/%/presentation.tex build/prepared
-	TEXINPUTS=./lib/cel-slides-template-2025:$$TEXINPUTS latexmk $<
-	mv build/presentation.pdf $@
+	TEXINPUTS=./lib/cel-slides-template-2025:$(dir $<):$$TEXINPUTS \
+		latexmk -outdir=build/$* $<
+	cp build/$*/presentation.pdf $@
 
 build/presentation_%_handout.pdf: src/%/presentation.tex build/prepared
-	TEXINPUTS=./lib/cel-slides-template-2025:$$TEXINPUTS latexmk -pdflatex='pdflatex %O "\def\ishandout{1}\input{%S}"' $<
-	mv build/presentation.pdf $@
+	TEXINPUTS=./lib/cel-slides-template-2025:$(dir $<):$$TEXINPUTS \
+		latexmk -outdir=build/$*_handout \
+		-pdflatex='$(RC_PDFLATEX) %O "\def\ishandout{1}\input{%S}"' $<
+	cp build/$*_handout/presentation.pdf $@
 
 build/prepared:
-	mkdir -p build
+	mkdir build
 	touch build/prepared
 
 .PHONY: clean

src/2026-01-16/presentation.tex

@@ -30,7 +30,10 @@
 \usepackage{tikz}
 \usepackage{tikz-3dplot}
 \usetikzlibrary{spy, external, intersections, positioning}
-%\tikzexternalize[prefix=build/]
+
+\ifdefined\ishandout\else
+\tikzexternalize
+\fi
 
 \usepackage{pgfplots}
 \pgfplotsset{compat=newest}
@@ -139,7 +142,7 @@
     \end{gather*}
     \vspace*{-2mm}
     \begin{gather*}
-        P_X(k) = \frac{\lambda^k}{k!}e^{-\lambda} \\[2mm]
+        P_X(n) = \frac{\lambda^n}{n!}e^{-\lambda} \\[2mm]
         \phi_X(s) = \text{exp}\left(\lambda (e^{js} -1)\right)
     \end{gather*}
     \vspace*{-2mm}
@@ -160,7 +163,7 @@
             \vspace*{-6mm}
             \begin{gather*}
                 X \sim \text{Poisson}(\lambda) \\[3mm]
-                P_X(k) = \frac{\lambda^k \cdot e^{-\lambda}}{k!} \\[4mm]
+                P_X(n) = \frac{\lambda^n \cdot e^{-\lambda}}{n!} \\[4mm]
                 \phi_X(s) = \text{exp}\left(\lambda (e^{js} -1)\right)
             \end{gather*}
         \end{greenblock}
@@ -168,7 +171,7 @@
             \vspace*{-6mm}
             \begin{gather*}
                 \nsum_{k=0}^{n} \binom{n}{k}a^k b^{n-k} = (a+b)^n, \hspace{15mm}
-                \binom{n}{k} = \frac{n!}{(n-k!)k!}
+                \binom{n}{k} = \frac{n!}{(n-k)!k!}
             \end{gather*}
         \end{greenblock}
         \column{\kitthreecolumns}
@@ -228,11 +231,11 @@
             zweier Zufallsvariablen.
             \pause\begin{gather*}
                 X \sim \text{Poisson}(\lambda_1) \hspace{3mm}
-                    \Leftrightarrow \hspace{3mm} P_X(k)
-                    = \frac{\lambda_1^k \cdot e^{-\lambda_1}}{k!} \hspace{30mm}
+                    \Leftrightarrow \hspace{3mm} P_X(n)
+                    = \frac{\lambda_1^n \cdot e^{-\lambda_1}}{n!} \hspace{30mm}
                 Y \sim \text{Poisson}(\lambda_2) \hspace{3mm}
-                    \Leftrightarrow \hspace{3mm} P_Y(k)
-                    = \frac{\lambda_2^k \cdot e^{-\lambda_2}}{k!}
+                    \Leftrightarrow \hspace{3mm} P_Y(n)
+                    = \frac{\lambda_2^n \cdot e^{-\lambda_2}}{n!}
             \end{gather*}
             \pause\begin{align*}
                 P_Z(n) &= P_{X+Y}(n) = \nsum_{k=0}^{n} P_X(k)P_Y(n-k)
@@ -291,64 +294,6 @@
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{Theorie Wiederholung}
 
-\begin{frame}
-    \frametitle{Unabhängigkeit \& Korrelation}
-
-    \vspace*{-10mm}
-
-    \begin{itemize}
-        \item Unabhängige ZV (stetig)
-            \begin{columns}
-                \column{\kitthreecolumns}
-                \begin{align*}
-                    X,Y \text{ unabhängig}
-                    \hspace{5mm} \Leftrightarrow \hspace{5mm}
-                    f_{X,Y}(x,y) = f_X(x)f_Y(y)
-                \end{align*}
-                \column{\kitthreecolumns}
-                \begin{lightgrayhighlightbox}
-                    Erinnerung: Unabhängige Ereignisse
-                    \begin{align*}
-                        X,Y \text{ \normalfont unabhängig}
-                        \hspace{5mm} \Leftrightarrow \hspace{5mm}
-                        P(AB) = P(A)P(B)
-                    \end{align*}
-                    \vspace*{-13mm}
-                \end{lightgrayhighlightbox}
-            \end{columns}
-            \pause
-        \item Kovarianz
-            \begin{columns}
-                \column{\kitthreecolumns}
-                \begin{align*}
-                    \text{cov}(X,Y) &= E\bigg( \big(X - E(X)\big) \big(Y
-                    - E(Y)\big) \bigg) \\
-                    &= E(XY) - E(X)E(Y)
-                \end{align*}
-                \column{\kitthreecolumns}
-                \begin{lightgrayhighlightbox}
-                    Erinnerung: Varianz
-                    \begin{align*}
-                        V(X) = E\big( \left(X - E(X)\right)^2 \big) = E(X^2) - E^2(X)
-                    \end{align*}
-                    \vspace*{-13mm}
-                \end{lightgrayhighlightbox}
-            \end{columns}
-        \item Korrelation
-            \begin{align*}
-                E(XY)
-            \end{align*}
-            \pause
-        \item Korrelationskoeffizient
-            \begin{align*}
-                \rho_{XY} = \frac{\text{cov}(X,Y)}{\sqrt{V(X)V(Y)}}
-                \hspace{25mm} \rho_{XY} = 0
-                \hspace{2mm}\Leftrightarrow\hspace{2mm}
-                E(XY) = E(X)E(Y)
-            \end{align*}
-    \end{itemize}
-\end{frame}
-
 \begin{frame}
     \frametitle{Mehrdimensionale Zufallsvariablen}
 
@@ -457,7 +402,66 @@
 \end{frame}
 
 \begin{frame}
-    \frametitle{Unabhängigkeit vs. Korrelation}
+    \frametitle{Unabhängigkeit \& Korrelation I}
+
+    \vspace*{-10mm}
+
+    \begin{itemize}
+        \item Unabhängige ZV (stetig)
+            \begin{columns}
+                \column{\kitthreecolumns}
+                \begin{align*}
+                    X,Y \text{ unabhängig}
+                    \hspace{5mm} \Leftrightarrow \hspace{5mm}
+                    f_{X,Y}(x,y) = f_X(x)f_Y(y)
+                \end{align*}
+                \column{\kitthreecolumns}
+                \begin{lightgrayhighlightbox}
+                    Erinnerung: Unabhängige Ereignisse
+                    \begin{align*}
+                        A,B \text{ \normalfont unabhängig}
+                        \hspace{5mm} \Leftrightarrow \hspace{5mm}
+                        P(AB) = P(A)P(B)
+                    \end{align*}
+                    \vspace*{-13mm}
+                \end{lightgrayhighlightbox}
+            \end{columns}
+            \pause
+        \item Kovarianz
+            \begin{columns}
+                \column{\kitthreecolumns}
+                \begin{align*}
+                    \text{cov}(X,Y) &= E\bigg( \big(X - E(X)\big) \big(Y
+                    - E(Y)\big) \bigg) \\
+                    &= E(XY) - E(X)E(Y)
+                \end{align*}
+                \column{\kitthreecolumns}
+                \begin{lightgrayhighlightbox}
+                    Erinnerung: Varianz
+                    \begin{align*}
+                        V(X) = E\big( \left(X - E(X)\right)^2 \big) =
+                        E(X^2) - E^2(X)
+                    \end{align*}
+                    \vspace*{-13mm}
+                \end{lightgrayhighlightbox}
+            \end{columns}
+        \item Korrelation
+            \begin{align*}
+                E(XY)
+            \end{align*}
+            \pause
+        \item Korrelationskoeffizient
+            \begin{align*}
+                \rho_{XY} = \frac{\text{cov}(X,Y)}{\sqrt{V(X)V(Y)}}
+                \hspace{25mm} \rho_{XY} = 0
+                \hspace{2mm}\Leftrightarrow\hspace{2mm}
+                E(XY) = E(X)E(Y)
+            \end{align*}
+    \end{itemize}
+\end{frame}
+
+\begin{frame}
+    \frametitle{Unabhängigkeit \& Korrelation II}
 
     \vspace*{-15mm}