From 081cad7f11838889e22ec76dac996215a07be0f0 Mon Sep 17 00:00:00 2001
From: Andreas Tsouchlos <an.tsouchlos@gmail.com>
Date: Sat, 17 Jan 2026 17:55:45 +0100
Subject: [PATCH] Fix factorial symbol and change variable k to n

---
 src/2026-01-16/presentation.tex | 19 ++++++++++---------
 1 file changed, 10 insertions(+), 9 deletions(-)

diff --git a/src/2026-01-16/presentation.tex b/src/2026-01-16/presentation.tex
index ece8135..e2f64be 100644
--- a/src/2026-01-16/presentation.tex
+++ b/src/2026-01-16/presentation.tex
@@ -142,7 +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}
@@ -163,7 +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}
@@ -171,7 +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}
@@ -231,11 +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)
@@ -419,7 +419,7 @@
                 \begin{lightgrayhighlightbox}
                     Erinnerung: Unabhängige Ereignisse
                     \begin{align*}
-                        X,Y \text{ \normalfont unabhängig}
+                        A,B \text{ \normalfont unabhängig}
                         \hspace{5mm} \Leftrightarrow \hspace{5mm}
                         P(AB) = P(A)P(B)
                     \end{align*}
@@ -439,7 +439,8 @@
                 \begin{lightgrayhighlightbox}
                     Erinnerung: Varianz
                     \begin{align*}
-                        V(X) = E\big( \left(X - E(X)\right)^2 \big) = E(X^2) - E^2(X)
+                        V(X) = E\big( \left(X - E(X)\right)^2 \big) =
+                        E(X^2) - E^2(X)
                     \end{align*}
                     \vspace*{-13mm}
                 \end{lightgrayhighlightbox}