From 20056bac47014caa3b44cab56d0a22e506ad4220 Mon Sep 17 00:00:00 2001
From: Andreas Tsouchlos <an.tsouchlos@gmail.com>
Date: Mon, 3 Nov 2025 13:34:47 +0100
Subject: [PATCH] tut3: Add theory 1 summary

---
 src/2025-12-05/presentation.tex | 66 ++++++++++++++++-----------------
 1 file changed, 31 insertions(+), 35 deletions(-)

diff --git a/src/2025-12-05/presentation.tex b/src/2025-12-05/presentation.tex
index 6a0e7a4..4bda3e4 100644
--- a/src/2025-12-05/presentation.tex
+++ b/src/2025-12-05/presentation.tex
@@ -291,7 +291,8 @@
         \centering
         \textbf{Poisson Verteilung}\\
         \vspace*{10mm}
-        Binomialverteilung für $N\rightarrow \infty$ mit $pN=\text{const.}=: \lambda$
+        Binomialverteilung für $N\rightarrow \infty$ mit
+        $pN=\text{const.}=: \lambda$
         \rule{0.9\textwidth}{0.4pt}
         \begin{gather*}
             X \sim \text{Poisson}(\lambda)
@@ -310,45 +311,40 @@
     \frametitle{Zusammenfassung}
 
     \begin{columns}
-        \column{\kitthreecolumns}
+        \column{\kittwocolumns}
+        \begin{greenblock}{Verteilungsfunktion (diskret)}
+            \vspace*{-6mm}
+            \begin{gather*}
+                F_X(x) = P(X \le x) = \sum_{n:x_n < x} P_X(x_n)
+            \end{gather*}
+        \end{greenblock}
+        \column{\kittwocolumns}
+        \begin{greenblock}{Erwartungswert}
+            \vspace*{-6mm}
+            \begin{gather*}
+                E(X) = \sum_{n=1}^{\infty} x_n P(X=x_n)
+            \end{gather*}%
+            \vspace*{-8mm}%
+            \begin{align*}
+                E(X + b) &= E(X) + b\\
+                E(X+Y) &= E(X) + E(Y)\\
+                E(aX) &= aE(X)
+            \end{align*}
+        \end{greenblock}
+        \column{\kittwocolumns}
         \begin{greenblock}{Binomialverteilung}
-            adsf
+            \vspace*{-6mm}
+            \begin{gather*}
+                P_X(k) = \binom{N}{k} p^k (1-p)^{1-k}
+            \end{gather*}
+            \begin{align*}
+                E(X) &= Np\\
+                V(X) &= Np(1-p)
+            \end{align*}
         \end{greenblock}
     \end{columns}
 \end{frame}
 
-% \begin{frame}
-%     \frametitle{Zusammenfassung}
-%
-%     \begin{columns}
-%         \column{\kitthreecolumns}
-%         \begin{greenblock}{Bedingte Wahrscheinlichkeit}
-%             \vspace*{-6mm}
-%             \begin{gather*}
-%                 P(A\vert B) = \frac{P(AB)}{P(B)}
-%             \end{gather*}
-%         \end{greenblock}
-%         \column{\kitthreecolumns}
-%         \begin{greenblock}{Formel von Bayes}
-%             \vspace*{-6mm}
-%             \begin{gather*}
-%                 P(A\vert B) = \frac{P(B\vert A) P(A)}{P(B)}
-%             \end{gather*}
-%         \end{greenblock}
-%     \end{columns}
-%     \begin{columns}
-%         \column{\kitonecolumn}
-%         \column{\kitthreecolumns}
-%         \begin{greenblock}{Satz der totalen Wahrscheinlichkeit}
-%             \vspace*{-6mm}
-%             \begin{gather*}
-%                 P(B) = \sum_{n} P(B\vert A_n)P(A_n)
-%             \end{gather*}
-%         \end{greenblock}
-%         \column{\kitonecolumn}
-%     \end{columns}
-% \end{frame}
-
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{Aufgabe}