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