tut4: Add some formulas for theory on part 2; Add TODOs

src/2025-12-19/presentation.tex

@@ -124,6 +124,9 @@
 \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(...)
+% TODO: V(X) = ... (preparation for use of sigma^2 in part two)
 \begin{frame}
     \frametitle{TODO}
 
@@ -359,8 +362,50 @@
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{Theorie Wiederholung}
 
+% TODO: Write
+% TODO: Plot
+% TODO: Mention it appears regularly because of the CLT
 \begin{frame}
-    \frametitle{sasdf}
+    \frametitle{Die Normalverteilung}
+
+    \begin{itemize}
+        \item TODO
+            \begin{align*}
+                f_X(x) = \frac{1}{\sqrt{2\pi \sigma^2}} \exp\left(\frac{(x - \mu)^2}{2 \sigma^2} \right) \\
+            \end{align*}
+    \end{itemize}
+\end{frame}
+
+% TODO: Write
+% TODO: Define Phi
+% TODO: Phi Rechenregeln
+% TODO: Define Explain use of tables
+\begin{frame}
+    \frametitle{Rechnen mithilfe der Standardnormalverteilung}
+
+    \begin{itemize}
+        \item Standardisierte ZV:
+            \begin{align*}
+                E(X) &= 0 \\
+                V(X) &= 1
+            \end{align*}
+        \item Standardisierung einer ZV:
+            \begin{align*}
+                \widetilde{X} = \frac{X - E(X)}{\sqrt{V(X)}}
+            \end{align*}
+        \item TODO:
+            \begin{gather*}
+                X \sim \mathcal{N}(\mu, \sigma^2) \\[5mm]
+                P(X \le a) = P\bigg(\underbrace{\frac{X - \mu}{\sigma}}_{:= Z \sim \mathcal{N}(0,1)} \le \frac{a - \mu}{\sigma}\bigg)
+                = P\bigg(Z \le \frac{a - \mu}{\sigma}\bigg) = \Phi\mleft( \frac{a - \mu}{\sigma}  \mright)
+            \end{gather*}
+    \end{itemize}
+\end{frame}
+
+% TODO: Write
+% TODO: Include Phi table?
+\begin{frame}
+    \frametitle{Zusammenfassung}
 \end{frame}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%