Add gen_histogram.py and finish theory for exercise 2
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src/2026-01-30/gen_histogram.py
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40
src/2026-01-30/gen_histogram.py
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import argparse
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import numpy as np
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import matplotlib.pyplot as plt
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from scipy.special import binom
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def array_to_pgfplots_table_string(a):
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return " ".join([f"({k}, {val})" for (k, val) in enumerate(a)]) + f" ({len(a)}, 0)"
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def P_binom(N, p, k):
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return binom(N, k) * p**k * (1 - p) ** (N - k)
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def main():
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# Parse command line arguments
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parser = argparse.ArgumentParser()
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parser.add_argument("-N", type=np.int32, required=True)
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parser.add_argument("-p", type=np.float32, required=True)
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parser.add_argument("--show", "-s", action="store_true")
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args = parser.parse_args()
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# Generate and show data
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N = args.N
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p = args.p
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bars = np.array([P_binom(N, p, k) for k in range(N + 1)])
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print(array_to_pgfplots_table_string(bars))
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if args.show:
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plt.stem(bars)
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plt.show()
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if __name__ == "__main__":
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main()
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@ -669,7 +669,80 @@
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\begin{frame}
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\begin{frame}
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\frametitle{Erinnerung: Rechnen mit Normalverteilungen}
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\frametitle{Erinnerung: Rechnen mit Normalverteilungen}
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% TODO: Write
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\vspace*{-21mm}
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\begin{itemize}
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\item Die Normalverteilung
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\end{itemize}
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\vspace*{-5mm}
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\begin{gather*}
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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)
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\hspace{20mm}
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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{gather*}
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\vspace*{-2mm}
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\begin{itemize}
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\item Die Standardnormalverteilung
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\end{itemize}
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\vspace{-5mm}
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\begin{minipage}{0.48\textwidth}
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\centering
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\begin{gather*}
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X \sim \mathcal{N} (0,1) \\[4mm]
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\Phi(x) := F_X(x) = P(X \le x) \\
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\Phi(-x) = 1 - \Phi(x)
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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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$x$ & $\Phi(x)$ & $x$ & $\Phi(x)$ & $x$ & $\Phi(x)$ \\
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\hline
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\hline
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$0{,}00$ & $0{,}500000$ & $0{,}10$ & $0{,}539828$ &
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$0{,}20$ & $0{,}579260$ \\
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$0{,}02$ & $0{,}507978$ & $0{,}12$ & $0{,}547758$ &
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$0{,}22$ & $0{,}587064$ \\
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$0{,}04$ & $0{,}515953$ & $0{,}14$ & $0{,}555670$ &
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$0{,}24$ & $0{,}594835$ \\
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$0{,}06$ & $0{,}523922$ & $0{,}16$ & $0{,}563559$ &
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$0{,}26$ & $0{,}602568$ \\
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$0{,}08$ & $0{,}531881$ & $0{,}18$ & $0{,}571424$ &
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$0{,}28$ & $0{,}610261$ \\
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\hline
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\end{tabular}\\
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\end{minipage}
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\begin{itemize}
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\item Standardisierung einer ZV
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\vspace*{-2mm}
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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{itemize}
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\vspace*{1mm}
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\begin{lightgrayhighlightbox}
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\vspace{-4mm}
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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\big(\underbrace{\widetilde{X}}_{\sim
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\mathcal{N}(0,1)} \le 0{,}24\big)
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= \Phi\left(0{,}24\right) = 0{,}594835
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\end{gather*}
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\vspace{-10mm}
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\end{lightgrayhighlightbox}
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\end{frame}
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\end{frame}
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\begin{frame}
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\begin{frame}
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@ -790,6 +863,7 @@
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\end{itemize}
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\end{itemize}
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\pause
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\pause
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\vspace*{2mm}
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\begin{minipage}[t]{0.32\textwidth}
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\begin{minipage}[t]{0.32\textwidth}
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\centering
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\centering
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\begin{figure}[H]
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\begin{figure}[H]
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@ -799,74 +873,41 @@
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\begin{axis}[
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\begin{axis}[
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width=10cm,
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width=10cm,
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height=5cm,
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height=5cm,
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scatter/classes={
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a={mark=*, blue}
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},
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xtick=\empty,
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xtick=\empty,
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ytick=\empty,
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ytick=\empty,
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xlabel = $x$,
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xlabel = $k$,
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ylabel = $y$,
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ylabel = $P_{S_N}(k)$,
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xmin=-4,xmax=4,
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area style,
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ymin=-1.5,ymax=2,
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]
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]
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\addplot+[
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\addplot+[scol0,fill=scol3,ybar interval,mark=no]
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scol1,
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plot coordinates
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scatter,
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{ (0,0.125) (1,0.375) (2,0.375) (3,0.125) (4,0) };
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only marks,
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\end{axis}
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scatter src=explicit symbolic,
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\end{tikzpicture}
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\end{figure}
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\end{minipage}%
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\begin{minipage}[t]{0.32\textwidth}
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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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width=10cm,
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height=5cm,
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xtick=\empty,
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ytick=\empty,
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xlabel = $k$,
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ylabel = $P_{S_N}(k)$,
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area style,
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]
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]
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table[row sep=crcr] {
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\addplot+[scol0,fill=scol3,ybar interval,mark=no]
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x y \\
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plot coordinates
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0.9782846466992505 1.3425401677691273 \\
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{
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-0.3342085827306991 -0.3478699656733771 \\
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(0,0.0009765625) (1,0.009765625) (2,0.0439453125)
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1.0329768177464096 0.906099042791728 \\
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(3,0.1171875) (4,0.205078125) (5,0.246 09375)
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0.4032837175133078 0.09609805659133519 \\
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(6,0.205078125) (7,0.1171875) (8,0.0439453125)
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-0.47995152749835157 -0.5885801242458046 \\
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(9,0.009765625) (10,0.0009765625) (11,0)
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-0.39301528503877914 0.5165601264867574 \\
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-0.3016076234682761 0.3555224809310629 \\
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-1.283841439924361 -1.092505952596916 \\
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0.6394093134607625 0.7760543139022245 \\
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-1.3930746204117168 -1.2539179604346171 \\
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0.7066349809976303 0.26736104561273705 \\
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-0.32358511023766134 -0.4974120460927544 \\
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0.5697159086054595 0.7427982778218153 \\
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0.7810330322454977 1.021722205669364 \\
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-1.05027750818351 -1.088249765156553 \\
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-0.18753992607010203 -0.4808932985122092 \\
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0.9163016000620543 1.1130761981874584 \\
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-0.16588501836421943 -0.5254281720340348 \\
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1.7319708376031673 1.2174504869365954 \\
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-0.5732884092151935 -0.4923142548003758 \\
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1.2626814172655978 1.1468156532099922 \\
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-1.1007514357735002 -0.626200459957605 \\
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0.40631320003662 -0.3705703698506922 \\
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-2.221684738838144 -2.739364284431091 \\
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1.1309626619949467 1.1940429603335854 \\
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0.3055128861785891 0.529524240616076 \\
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-0.22522789028651527 -0.5082861632170081 \\
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0.2726524372676921 -0.2466404699684424 \\
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0.7078557266441373 0.8428284296154347 \\
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-1.4402649481540337 -0.9344326515164862 \\
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1.129522000340855 0.4510295424893529 \\
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0.5870764491195138 0.5669363454321612 \\
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2.3539677525351856 2.2253385575502285 \\
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0.2028654829519406 0.24539632425150296 \\
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-0.20861363707807395 -0.26125228812993867 \\
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0.6187802012217948 -0.2685299708916181 \\
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-0.2659232421672081 -0.22662166465228362 \\
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1.2403675794143405 1.0157380953006032 \\
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-0.0391562905128538 -0.6304153520459441 \\
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0.9833408241524402 1.06523679491654 \\
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0.5231710701516994 0.44339120385526315 \\
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-3.2527645820047146 -2.955881198077996 \\
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0.08993024102635327 0.6534559407213543 \\
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0.4076640826339743 0.5075313685387366 \\
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0.7431965606838403 0.04225691288802064 \\
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2.0420226454403996 2.411788877111026 \\
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-0.3652226483952774 -0.5846718876921133 \\
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0.478643720906727 0.7267990110235567 \\
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1.18457297115014 1.7548366300308906 \\
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0.05462743086401826 -0.02632310517996274\\
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};
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};
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\end{axis}
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\end{axis}
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\end{tikzpicture}
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\end{tikzpicture}
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@ -881,187 +922,152 @@
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\begin{axis}[
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\begin{axis}[
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width=10cm,
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width=10cm,
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height=5cm,
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height=5cm,
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scatter/classes={
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a={mark=*, blue}
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},
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xtick=\empty,
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xtick=\empty,
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ytick=\empty,
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ytick=\empty,
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xlabel = $x$,
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xlabel = $k$,
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ylabel = $y$,
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ylabel = $P_{S_N}(k)$,
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xmin=-2,xmax=1.7,
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area style,
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ymin=-6,ymax=6,
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]
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]
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\addplot+[
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\addplot+[scol0,fill=scol3,ybar interval,mark=no]
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scol1,
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plot coordinates
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scatter,
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{
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only marks,
|
(0,8.881784197001252e-16)
|
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scatter src=explicit symbolic,
|
(1,4.440892098500626e-14)
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]
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(2,1.0880185641326534e-12)
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table[row sep=crcr] {
|
(3,1.7408297026122455e-11)
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x y \\
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(4,2.0454749005693884e-10)
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1.474648977967909 1.4746489779679088 \\
|
(5,1.8818369085238373e-09)
|
||||||
0.16441809142886987 0.16441809142886984 \\
|
(6,1.411377681392878e -08)
|
||||||
-2.031822997202981 -2.0318229972029807 \\
|
(7,8.871516854469519e-08)
|
||||||
1.2182520939353458 1.2182520939353456 \\
|
(8,4.768440309277366e-07)
|
||||||
-0.5531479118291289 -0.5531479118291288 \\
|
(9,2.2252721443294377e-06)
|
||||||
-1.47499216570319 -1.4749921657031897 \\
|
(10,9.123615791750694e-06)
|
||||||
0.6306758881059998 0.6306758881059997 \\
|
(11,3.317678469727525e-05)
|
||||||
-1.5640176651059605 -1.56401766510596 \\
|
(12,0.00010782455026614456)
|
||||||
-1.3859450407248939 -1.3859450407248937 \\
|
(13,0.00031517945462411495)
|
||||||
-1.0665944205353617 -1.0665944205353615 \\
|
(14,0.0008329742729351608)
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||||||
0.7492473817601838 0.7492473817601837 \\
|
(15,0.001999138255044386)
|
||||||
0.9499783172548493 0.949978317254849 \\
|
(16,0.004373114932909594)
|
||||||
0.34018738058932396 0.3401873805893239 \\
|
(17,0.00874622986581919)
|
||||||
-1.0096550061469416 -1.0096550061469414 \\
|
(18,0.016034754754001845)
|
||||||
-0.6754142249990085 -0.6754142249990084 \\
|
(19,0.02700590274358206)
|
||||||
-0.7514107790472649 -0.7514107790472647 \\
|
(20,0.04185914925255218)
|
||||||
0.7509769467458492 0.750976946745849 \\
|
(21,0.059798784646503136)
|
||||||
0.646133492401253 0.6461334924012528 \\
|
(22,0.0788256706703905)
|
||||||
-1.745989051249313 -1.7459890512493126 \\
|
(23,0.09596168603351884)
|
||||||
-0.29616542476720953 -0.2961654247672095 \\
|
(24,0.10795689678770869)
|
||||||
0.845676968386932 0.8456769683869317 \\
|
(25,0.11227517265921708)
|
||||||
-1.5863350761719144 -1.586335076171914 \\
|
(26,0.10795689678770869)
|
||||||
-0.42188175903423886 -0.42188175903423875 \\
|
(27,0.09596168603351884)
|
||||||
-1.6501793388980994 -1.650179338898099 \\
|
(28,0.0788256706703905)
|
||||||
-0.5605813083780707 -0.5605813083780706 \\
|
(29,0.059798784646503136)
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||||||
0.9976042516699373 0.997604251669937 \\
|
(30,0.04185914925255218)
|
||||||
-1.4071020277527053 -1.407102027752705 \\
|
(31,0.02700590274358206)
|
||||||
1.2028353996291026 1.2028353996291024 \\
|
(32,0.016034754754001845)
|
||||||
1.0123659351742527 1.0123659351742524 \\
|
(33,0.00874622986581919)
|
||||||
1.3398753611390357 1.3398753611390355 \\
|
(34,0.004373114932909594)
|
||||||
-1.8353880184343434 -1.835388018434343 \\
|
(35,0.001999138255044386)
|
||||||
0.20942595245430015 0.2094259524543001 \\
|
(36,0.0008329742729351608)
|
||||||
-0.8821105204722243 -0.882110520472224 \\
|
(37,0.00031517945462411495)
|
||||||
0.2823899445624717 0.28238994456247163 \\
|
(38,0.00010782455026614456)
|
||||||
-0.6518146330435214 -0.6518146330435213 \\
|
(39,3.317678469727525e-05)
|
||||||
0.48917046669684383 0.4891704666968437 \\
|
(40,9.123615791750694e-06)
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||||||
0.8356612510810297 0.8356612510810295 \\
|
(41,2.2252721443294377e-06)
|
||||||
-0.33074175460821187 -0.3307417546082118 \\
|
(42,4.768440309277366e-07)
|
||||||
0.2947454674362563 0.29474546743625624 \\
|
(43,8.871516854469519e-08)
|
||||||
-1.2159944359594155 -1.2159944359594153 \\
|
(44,1.411377681392878e-08)
|
||||||
-0.21681449173721903 -0.21681449173721898 \\
|
(45,1.8818369085238373e-09)
|
||||||
-0.7152726760226845 -0.7152726760226844 \\
|
(46,2.0454749005693884e-10)
|
||||||
1.3162334939362095 1.3162334939362093 \\
|
(47,1.7408297026122455e-11)
|
||||||
0.2527835241858269 0.25278352418582684 \\
|
(48,1.0880185641326534e-12)
|
||||||
-0.9986661858536788 -0.9986661858536786 \\
|
(49,4.440892098500626e-14)
|
||||||
-0.9568391971929098 -0.9568391971929096 \\
|
(50,8.881784197001252e-16)
|
||||||
-0.5474319285933091 -0.547431928593309 \\
|
(51,0)
|
||||||
-2.30328755152605 -2.3032875515260494 \\
|
|
||||||
-1.2929155901531633 -1.292915590153163 \\
|
|
||||||
-0.8530475462391427 -0.8530475462391425\\
|
|
||||||
};
|
|
||||||
\end{axis}
|
|
||||||
\end{tikzpicture}
|
|
||||||
\end{figure}
|
|
||||||
\end{minipage}%
|
|
||||||
\begin{minipage}[t]{0.32\textwidth}
|
|
||||||
\centering
|
|
||||||
\begin{figure}[H]
|
|
||||||
\centering
|
|
||||||
|
|
||||||
\begin{tikzpicture}
|
|
||||||
\begin{axis}[
|
|
||||||
width=10cm,
|
|
||||||
height=5cm,
|
|
||||||
scatter/classes={
|
|
||||||
a={mark=*, blue}
|
|
||||||
},
|
|
||||||
xtick=\empty,
|
|
||||||
ytick=\empty,
|
|
||||||
xlabel = $x$,
|
|
||||||
ylabel = $y$,
|
|
||||||
xmin=-4,xmax=4,
|
|
||||||
ymin=-2,ymax=2,
|
|
||||||
]
|
|
||||||
\addplot+[
|
|
||||||
scol1,
|
|
||||||
scatter,
|
|
||||||
only marks,
|
|
||||||
scatter src=explicit symbolic,
|
|
||||||
]
|
|
||||||
table[row sep=crcr] {
|
|
||||||
x y \\
|
|
||||||
-1.7077500965534018 0.9715072286946655 \\
|
|
||||||
1.1148806736392152 0.9230117606631614 \\
|
|
||||||
0.11058932943085453 -0.5157596318522968 \\
|
|
||||||
0.08527262614233909 -0.9720863462538933 \\
|
|
||||||
-1.4204389641047823 -0.9712150414232805 \\
|
|
||||||
-0.6705061079694784 -0.061860055599544606 \\
|
|
||||||
-0.6212830814536863 1.2589504540208847 \\
|
|
||||||
1.4236240086652356 -1.302789472184279 \\
|
|
||||||
-1.0975355477486632 -0.886909899437918 \\
|
|
||||||
1.5752231220795536 1.2260114881873 \\
|
|
||||||
1.2049717160174165 1.0705757620706944 \\
|
|
||||||
-1.7929521084203113 -1.0124364432205855 \\
|
|
||||||
1.1345482934601252 -0.7213210134187505 \\
|
|
||||||
0.06993810174580865 -1.5278087661910722 \\
|
|
||||||
0.50560442840041 1.1191719084519776 \\
|
|
||||||
-0.814167507403749 0.2021470144855546 \\
|
|
||||||
2.03061011925002 0.08990067866176893 \\
|
|
||||||
0.7257818062658367 0.22602273591014058 \\
|
|
||||||
0.5036942935085902 0.2520250465804246 \\
|
|
||||||
0.5973644458579076 -0.2093760967114109 \\
|
|
||||||
1.1104283164930224 1.5071527221448955 \\
|
|
||||||
-0.052216510646198096 -0.5465573566030532 \\
|
|
||||||
0.423205976943666 -0.21077815853809784 \\
|
|
||||||
0.2982451040844636 -1.3591258564459687 \\
|
|
||||||
0.539438662504297 -0.780387830281188 \\
|
|
||||||
0.08417174139937453 0.2725275842632153 \\
|
|
||||||
0.05733773656028022 0.8226842222044897 \\
|
|
||||||
0.12184004421107687 -1.0962860273484687 \\
|
|
||||||
3.0973129011059326 0.13325075656192403 \\
|
|
||||||
1.464718817591499 -2.0541680373660234 \\
|
|
||||||
0.6017327837974983 -0.43330515099025413 \\
|
|
||||||
-1.6527036180073127 -0.04153499563379528 \\
|
|
||||||
1.3583641617521591 -0.9127837751641491 \\
|
|
||||||
-0.2808122864213532 0.6566355071818034 \\
|
|
||||||
0.36085503878766245 -0.2372816111687184 \\
|
|
||||||
-0.7808961491915221 -0.4569496546349541 \\
|
|
||||||
-0.08144830754364803 0.5297194167082963 \\
|
|
||||||
-0.3832453043478111 0.695158762430314 \\
|
|
||||||
-0.3021005547959829 -0.7515146005101381 \\
|
|
||||||
0.0832540012145203 -1.6257847886861803 \\
|
|
||||||
-0.08783078629061673 0.48401963778829576 \\
|
|
||||||
0.5098330610876248 0.3327688893197499 \\
|
|
||||||
0.4804292632122983 -0.5397408326625166 \\
|
|
||||||
0.3612454424603153 -0.2728088913965057 \\
|
|
||||||
0.8706855868841972 1.8337909595106936 \\
|
|
||||||
-0.7868151662161218 1.643221471861054 \\
|
|
||||||
-0.5629480754112661 0.16190044666568626 \\
|
|
||||||
0.9623486507086952 -0.06821392925238735 \\
|
|
||||||
-0.390445497949156 -1.4902360360777431 \\
|
|
||||||
2.239228377147278 -0.2037307482272916\\
|
|
||||||
};
|
};
|
||||||
\end{axis}
|
\end{axis}
|
||||||
\end{tikzpicture}
|
\end{tikzpicture}
|
||||||
\end{figure}
|
\end{figure}
|
||||||
\end{minipage}%
|
\end{minipage}%
|
||||||
|
|
||||||
\vspace*{3mm}
|
\vspace*{-1mm}
|
||||||
|
|
||||||
\begin{minipage}[t]{0.32\textwidth}
|
\begin{minipage}[t]{0.32\textwidth}
|
||||||
\centering
|
\centering
|
||||||
$N = 10$
|
$N = 4, p=0{,}5$
|
||||||
\end{minipage}%
|
\end{minipage}%
|
||||||
\begin{minipage}[t]{0.32\textwidth}
|
\begin{minipage}[t]{0.32\textwidth}
|
||||||
\centering
|
\centering
|
||||||
$N = 100$
|
$N = 10, p=0{,}5$
|
||||||
\end{minipage}%
|
\end{minipage}%
|
||||||
\begin{minipage}[t]{0.32\textwidth}
|
\begin{minipage}[t]{0.32\textwidth}
|
||||||
\centering
|
\centering
|
||||||
$N = 1000$
|
$N = 50, p=0{,}5$
|
||||||
\end{minipage}%
|
\end{minipage}%
|
||||||
\end{frame}
|
\end{frame}
|
||||||
|
|
||||||
\begin{frame}
|
\begin{frame}
|
||||||
\frametitle{Zusammenfassung}
|
\frametitle{Zusammenfassung}
|
||||||
|
|
||||||
\begin{itemize}
|
\vspace*{-25mm}
|
||||||
\item Tabelle
|
|
||||||
\item $\Phi(-x) = 1 - \Phi(x)$
|
|
||||||
\item ZGWS: Approx von Binom.
|
|
||||||
\end{itemize}
|
|
||||||
|
|
||||||
% TODO: Write
|
\begin{columns}[t]
|
||||||
|
\column{\kitthreecolumns}
|
||||||
|
\centering
|
||||||
|
\begin{greenblock}{Standardnormalverteilung}
|
||||||
|
\vspace*{-10mm}
|
||||||
|
\begin{gather*}
|
||||||
|
X \sim \mathcal{N} (0,1) \\[4mm]
|
||||||
|
\Phi(x) := F_X(x) = P(X \le x) \\
|
||||||
|
\Phi(-x) = 1 - \Phi(x)
|
||||||
|
\end{gather*}
|
||||||
|
\end{greenblock}
|
||||||
|
\begin{greenblock}{Standardisierung}
|
||||||
|
\vspace*{-10mm}
|
||||||
|
\begin{gather*}
|
||||||
|
\widetilde{X} = \frac{X - E(X)}{\sqrt{V(X)}}
|
||||||
|
= \frac{X - \mu}{\sigma}
|
||||||
|
\end{gather*}
|
||||||
|
\end{greenblock}
|
||||||
|
\column{\kitthreecolumns}
|
||||||
|
\centering
|
||||||
|
\begin{greenblock}{Approximation einer Binom.vert. mit dem ZGWS}
|
||||||
|
\vspace*{-10mm}
|
||||||
|
\begin{gather*}
|
||||||
|
\text{Bedingung: } Np(1-p) \ge 9
|
||||||
|
\end{gather*}
|
||||||
|
\vspace*{-7mm}
|
||||||
|
\begin{align*}
|
||||||
|
P_X(a < S_N \le b) &= \nsum_{k=a}^{b} \binom{N}{k}
|
||||||
|
p^k(1-p)^{N-k} \\
|
||||||
|
& \approx
|
||||||
|
\Phi\left(\frac{b - Np}{\sqrt{Np(1-p)}}\right) -
|
||||||
|
\Phi\left(\frac{a - Np}{\sqrt{Np(1-p)}}\right)
|
||||||
|
\end{align*}
|
||||||
|
\end{greenblock}
|
||||||
|
\end{columns}
|
||||||
|
|
||||||
|
\vspace*{5mm}
|
||||||
|
|
||||||
|
\begin{table}
|
||||||
|
\centering
|
||||||
|
\begin{tabular}{|c|c||c|c||c|c|}
|
||||||
|
\hline
|
||||||
|
$x$ & $\Phi(x)$ & $x$ & $\Phi(x)$ & $x$ & $\Phi(x)$ \\
|
||||||
|
\hline
|
||||||
|
\hline
|
||||||
|
$1{,}60$ & $0{,}945201$ & $2{,}00$ & $0{,}977250$ &
|
||||||
|
$2{,}40$ & $0{,}991802$ \\
|
||||||
|
$1{,}62$ & $0{,}947384$ & $2{,}02$ & $0{,}978308$ &
|
||||||
|
$2{,}42$ & $0{,}992240$ \\
|
||||||
|
$1{,}64$ & $0{,}949497$ & $2{,}04$ & $0{,}979325$ &
|
||||||
|
$2{,}44$ & $0{,}992656$ \\
|
||||||
|
$1{,}66$ & $0{,}951543$ & $2{,}06$ & $0{,}980301$ &
|
||||||
|
$2{,}46$ & $0{,}993053$ \\
|
||||||
|
$1{,}68$ & $0{,}953521$ & $2{,}08$ & $0{,}981237$ &
|
||||||
|
$2{,}48$ & $0{,}993431$ \\
|
||||||
|
\hline
|
||||||
|
\end{tabular}
|
||||||
|
\end{table}
|
||||||
\end{frame}
|
\end{frame}
|
||||||
|
|
||||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||||
@ -1072,8 +1078,8 @@
|
|||||||
|
|
||||||
Im Werk einer Zahnradfabrik werden verschiedene
|
Im Werk einer Zahnradfabrik werden verschiedene
|
||||||
Präzisionsmetallteile gefertigt. Während einer
|
Präzisionsmetallteile gefertigt. Während einer
|
||||||
Schicht werden 5000 Stück eines Typs A hergestellt. Bei der
|
Schicht werden $5000$ Stück eines Typs A hergestellt. Bei der
|
||||||
Qualitätskontrolle werden 3% dieser
|
Qualitätskontrolle werden $3\%$ dieser
|
||||||
Teile als defekt klassifiziert und aussortiert.
|
Teile als defekt klassifiziert und aussortiert.
|
||||||
|
|
||||||
% tex-fmt: off
|
% tex-fmt: off
|
||||||
|
|||||||
Loading…
Reference in New Issue
Block a user