Add theory for part 2

src/2025-12-19/presentation.tex

@@ -386,50 +386,194 @@
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{Theorie Wiederholung}
 
-% TODO: Write
-% TODO: Plot
-% TODO: Mention it appears regularly because of the CLT
 \begin{frame}
     \frametitle{Die Normalverteilung}
 
-    \begin{itemize}
+    \begin{columns}
-        \item TODO
+        \column{\kitthreecolumns}
+        \centering
+        \begin{gather*}
+            X \sim \mathcal{N}\mleft( \mu, \sigma^2 \mright)
+        \end{gather*}%
+        \vspace{2mm}
         \begin{align*}
-                f_X(x) = \frac{1}{\sqrt{2\pi \sigma^2}} \exp\left(\frac{(x - \mu)^2}{2 \sigma^2} \right) \\
+            f_X(x) &= \frac{1}{\sqrt{2\pi \sigma^2}} \exp\left(\frac{(x -
+            \mu)^2}{2 \sigma^2} \right) \\[2mm]
+            F_X(x) &=
+            \vcenter{\hbox{\scalebox{1.5}[2.6]{\vspace*{3mm}$\displaystyle\int$}}}_{\hspace{-0.5em}-\infty}^{\,x}
+            \frac{1}{\sqrt{2\pi
+            \sigma^2}} \exp\left(\frac{(u - \mu)^2}{2 \sigma^2} \right) du
         \end{align*}
-    \end{itemize}
+        \column{\kitthreecolumns}
+        \centering
+        \begin{figure}[H]
+            \centering
+            \begin{tikzpicture}
+                \begin{axis}[
+                        domain=-4:4,
+                        xmin=-4,xmax=4,
+                        width=15cm,
+                        height=5cm,
+                        samples=200,
+                        xlabel={$x$},
+                        ylabel={$f_X(x)$},
+                        xtick={0},
+                        xticklabels={\textcolor{KITblue}{$\mu$}},
+                        ytick={0},
+                    ]
+                    \addplot+[mark=none, line width=1pt]
+                    {(1 / sqrt(2*pi)) * exp(-x*x)};
+
+                    \addplot+ [KITblue, mark=none, line width=1pt]
+                    coordinates {(-0.5, 0.15) (0.5, 0.15)};
+                    \addplot+ [KITblue, mark=none, line width=1pt]
+                    coordinates {(-0.5, 0.12) (-0.5, 0.18)};
+                    \addplot+ [KITblue, mark=none, line width=1pt]
+                    coordinates {(0.5, 0.12) (0.5, 0.18)};
+                    \node[KITblue] at (axis cs: 0, 0.2) {$\sigma$};
+
+                    % \addplot +[scol2, mark=none, line width=1pt]
+                    % coordinates {(4.8, -1) (4.8, 2)};
+                    % \addplot +[scol2, mark=none, line width=1pt]
+                    % coordinates {(5.2, -1) (5.2, 2)};
+                    % \node at (axis cs: 4.8, 3) {$S(1-\delta)$};
+                    % \node at (axis cs: 5.2, 3) {$S(1+\delta)$};
+                \end{axis}
+            \end{tikzpicture}
+        \end{figure}
+        \begin{figure}[H]
+            \centering
+            \begin{tikzpicture}
+                \begin{axis}[
+                        domain=-4:4,
+                        xmin=-4,xmax=4,
+                        width=15cm,
+                        height=5cm,
+                        samples=200,
+                        xlabel={$x$},
+                        ylabel={$F_X(x)$},
+                        xtick=\empty,
+                        ytick={0, 1},
+                    ]
+                    \addplot+[mark=none, line width=1pt]
+                    {1 / (1 + exp(-(1.526*x*(1 + 0.1034*x))))};
+                \end{axis}
+            \end{tikzpicture}
+        \end{figure}
+    \end{columns}
 \end{frame}
 
-% TODO: Write
+% TODO: Are Z/z notation used in the lecture?
-% TODO: Define Phi
-% TODO: Phi Rechenregeln
-% TODO: Define Explain use of tables
 \begin{frame}
     \frametitle{Rechnen mithilfe der Standardnormalverteilung}
 
+    \vspace*{-15mm}
+
     \begin{itemize}
-        \item Standardisierte ZV:
+        \item Die Standardnormalverteilung
-            \begin{align*}
+    \end{itemize}
+
+    \begin{minipage}{0.48\textwidth}
+        \centering
+        \begin{gather*}
+            Z \sim \mathcal{N} (0,1) \\[4mm]
+            \Phi(z) := F_Z(z) = P(Z \le z) \\
+            \Phi(-z) = 1 - \Phi(z)
+        \end{gather*}
+    \end{minipage}%
+    \begin{minipage}{0.48\textwidth}
+        \centering
+        \begin{tabular}{|c|c||c|c||c|c|}
+            \hline
+            $z$ & $\Phi(z)$ & $z$ & $\Phi(z)$ & $z$ & $\Phi(z)$ \\
+            \hline
+            \hline
+            0{,}00 & 0{,}500000 & 0{,}10 & 0{,}539828 & 0{,}20 & 0{,}579260 \\
+            0{,}02 & 0{,}507978 & 0{,}12 & 0{,}547758 & 0{,}22 & 0{,}587064 \\
+            0{,}04 & 0{,}515953 & 0{,}14 & 0{,}555670 & 0{,}24 & 0{,}594835 \\
+            0{,}06 & 0{,}523922 & 0{,}16 & 0{,}563559 & 0{,}26 & 0{,}602568 \\
+            0{,}08 & 0{,}531881 & 0{,}18 & 0{,}571424 & 0{,}28 & 0{,}610261 \\
+            \hline
+        \end{tabular}\\
+    \end{minipage}
+
+    \pause
+    \begin{itemize}
+        \item Standardisierte ZV
+            \begin{gather*}
+                \begin{array}{cc}
                     E(X) &= 0 \\
                     V(X) &= 1
-            \end{align*}
+                \end{array}
-        \item Standardisierung einer ZV:
+                \hspace{45mm}
-            \begin{align*}
+                \text{Standardisierung: } \hspace{5mm}
                 \widetilde{X} = \frac{X - E(X)}{\sqrt{V(X)}}
-            \end{align*}
+                = \frac{X - \mu}{\sigma}
-        \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}
+
+    \vspace*{5mm}
+
+    \pause
+    \begin{lightgrayhighlightbox}
+        Rechenbeispiel
+        \begin{gather*}
+            X \sim \mathcal{N}(\mu = 1, \sigma^2 = 0{,}5^2) \\[2mm]
+            P\left(X \le 1{,}12 \right)
+            = P\left(\frac{X - 1}{0{,}5} \le \frac{1{,}12 - 1}{0{,}5}\right)
+            = P\left(\frac{X - 1}{0{,}5} \le
+            0{,}24\right)  = \Phi\left(0{,}24\right) = 0{,}594835
+        \end{gather*}
+    \end{lightgrayhighlightbox}
 \end{frame}
 
-% TODO: Write
+% TODO: Are Z/z notation used in the lecture?
-% TODO: Include Phi table?
 \begin{frame}
     \frametitle{Zusammenfassung}
+
+    \vspace*{-15mm}
+
+    \begin{columns}[t]
+        \column{\kitthreecolumns}
+        \centering
+        \begin{greenblock}{Standardnormalverteilung}
+            \vspace*{-10mm}
+            \begin{gather*}
+                Z \sim \mathcal{N} (0,1) \\[4mm]
+                \Phi(z) := F_Z(z) = P(Z \le z) \\
+                \Phi(-z) = 1 - \Phi(z)
+            \end{gather*}
+        \end{greenblock}
+        \column{\kitthreecolumns}
+        \centering
+        \begin{greenblock}{Standardisierung}
+            \vspace*{-10mm}
+            \begin{gather*}
+                \widetilde{X} = \frac{X - E(X)}{\sqrt{V(X)}}
+                = \frac{X - \mu}{\sigma}
+            \end{gather*}
+        \end{greenblock}
+    \end{columns}
+
+    \vspace{5mm}
+
+    \begin{table}
+        \centering
+        % \cdots
+        \begin{tabular}{|c|c||c|c||c|c||c|c|}
+            \hline
+            $z$ & $\Phi(z)$ & $z$ & $\Phi(z)$ & $z$ & $\Phi(z)$ & $z$ & $\Phi(z)$ \\
+            \hline
+            \hline
+            1{,}40 & 0{,}919243 & 2{,}80 & 0{,}997445 & 3{,}00 & 0{,}998650 & 4{,}20 & 0{,}999987 \\
+            1{,}42 & 0{,}922196 & 2{,}82 & 0{,}997599 & 3{,}02 & 0{,}998736 & 4{,}22 & 0{,}999988 \\
+            1{,}44 & 0{,}925066 & 2{,}84 & 0{,}997744 & 3{,}04 & 0{,}998817 & 4{,}24 & 0{,}999989 \\
+            1{,}46 & 0{,}927855 & 2{,}86 & 0{,}997882 & 3{,}06 & 0{,}998893 & 4{,}26 & 0{,}999990 \\
+            1{,}48 & 0{,}930563 & 2{,}88 & 0{,}998012 & 3{,}08 & 0{,}998965 & 4{,}28 & 0{,}999991 \\
+            \hline
+        \end{tabular}
+        % \cdots
+    \end{table}
 \end{frame}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -481,7 +625,7 @@
             \column{\kitthreecolumns}
             \centering
             \pause \begin{gather*}
-                X \sim \mathcal{N} \mleft( \mu = 0{,}5, \sigma = 0{,}07^2 \mright)
+                X \sim \mathcal{N} \mleft( \mu = 0{,}5, \sigma^2 = 0{,}07^2 \mright)
             \end{gather*}
             \begin{align*}
                 P(E_\text{a}) &= P \Big( \big( X < S(1-\delta) \big)
@@ -511,8 +655,8 @@
                         \addplot+[mark=none, line width=1pt]
                             {1 / sqrt(2*3.1415*0.07*0.07) * exp(-(x - 5)*(x-5)/(2*0.07*0.07))};
 
-                        \addplot +[scol2, mark=none, line width=1pt] coordinates {(4.8, -1) (4.8, 2)};
+                        \addplot +[KITblue, mark=none, line width=1pt] coordinates {(4.8, -1) (4.8, 2)};
-                        \addplot +[scol2, mark=none, line width=1pt] coordinates {(5.2, -1) (5.2, 2)};
+                        \addplot +[KITblue, mark=none, line width=1pt] coordinates {(5.2, -1) (5.2, 2)};
                         \node at (axis cs: 4.8, 3) {$S(1-\delta)$};
                         \node at (axis cs: 5.2, 3) {$S(1+\delta)$};