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Add solution for exercise 2c

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Andreas Tsouchlos 2026-01-21 09:53:26 +01:00
parent 6098da86fa
commit dfc558ca16
1 changed files with 96 additions and 14 deletions
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src/2026-01-30/presentation.tex
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@ -351,27 +351,48 @@
% tex-fmt: on % tex-fmt: on
\vspace*{-5mm} \vspace*{-5mm}
\pause
\begin{gather*} \begin{gather*}
\tilde{S}_N \sim \mathcal{N}(\mu = 150, \sigma^2 = 145{,}5) \\ \tilde{S}_N \sim \mathcal{N}(\mu = 150, \sigma^2 = 145{,}5) \\
\end{gather*} \end{gather*}
\vspace*{-5mm} \pause
\pause\begin{align*} \begin{minipage}[t]{0.2\textwidth}
P(\tilde{S}_N \le dx) \ge 0{,}98 \phantom{placeholder}
\hspace*{5mm} &\Rightarrow \hspace*{5mm} \end{minipage}
P\left(\frac{\tilde{S}_N - E(\tilde{S}_N)}{\sqrt{V(\tilde{S}_N)}} \begin{minipage}[t]{0.2\textwidth}
\le \frac{dx - 150}{\sqrt{145{,}5}} \right) \ge 0{,}98 \\[3mm] \vspace*{-4mm}
&\Rightarrow \hspace*{5mm} \centering
\Phi\left( \frac{dx - 150}{\sqrt{145{,}5}} \right) \ge 0{,}98 \\[3mm] \begin{gather*}
&\Rightarrow \hspace*{5mm} P(\tilde{S}_N \le dx) \ge 0{,}98
\frac{dx - 150}{\sqrt{145{,}5}} \ge \Phi^{-1}(0{,}98) = 2{,}06 \\[3mm] \end{gather*}
&\Rightarrow \hspace*{5mm} \end{minipage}%
dx \ge 174{,}8 \pause
\end{align*} \begin{minipage}[t]{0.3\textwidth}
\vspace*{-15mm}
\begin{align*}
&\Rightarrow \hspace*{5mm}
P\Bigg(\overbrace{\frac{\tilde{S}_N -
E(\tilde{S}_N)}{\sqrt{V(\tilde{S}_N)}}}^{\sim \mathcal{N}(0,1)}
\le \frac{dx - 150}{\sqrt{145{,}5}} \Bigg) \ge 0{,}98
\end{align*}
\vspace*{-5mm}
\pause
\begin{align*}
&\Rightarrow \hspace*{5mm}
\Phi\left( \frac{dx - 150}{\sqrt{145{,}5}} \right) \ge
0{,}98 \\[3mm]
&\Rightarrow \hspace*{5mm}
\frac{dx - 150}{\sqrt{145{,}5}} \ge \Phi^{-1}(0{,}98) =
2{,}06 \\[3mm]
&\Rightarrow \hspace*{5mm}
dx \ge 174{,}8
\end{align*}
\end{minipage}
\pause
\centering \centering
\vspace*{10mm} \vspace*{10mm}
Der Kessel muss mindestens $175$ Teile fassen Der Kessel muss mindestens $175$ Teile fassen
\end{frame} \end{frame}
\begin{frame} \begin{frame}
@ -386,6 +407,67 @@
Wahrscheinlichkeit von $0{,}98$ nicht überfüllt ist? Wahrscheinlichkeit von $0{,}98$ nicht überfüllt ist?
\end{enumerate} \end{enumerate}
% tex-fmt: on % tex-fmt: on
\vspace*{-10mm}
\pause
\begin{gather*}
\tilde{S}_N \sim \mathcal{N}\left(\mu = Np, \sigma^2 = Np(1-p)\right) \\
\end{gather*}
\pause
\begin{minipage}[t]{0.2\textwidth}
\vspace*{-4mm}
\centering
\begin{gather*}
P(\tilde{S}_N \le 200) \ge 0{,}98
\end{gather*}
\end{minipage}%
\pause
\begin{minipage}[t]{0.3\textwidth}
\vspace*{-15mm}
\begin{align*}
&\Rightarrow \hspace*{5mm}
P\Bigg(\overbrace{\frac{\tilde{S}_N -
E(\tilde{S}_N)}{\sqrt{V(\tilde{S}_N)}}}^{\sim
\mathcal{N}(0,1)}
\le \frac{200 - Np}{\sqrt{Np(1-p)} } \Bigg) \ge 0{,}98
\end{align*}
\vspace*{-5mm}
\pause
\begin{align*}
&\Rightarrow \hspace*{5mm}
\Phi\left( \frac{200 - Np}{\sqrt{Np(1-p)}} \right) \ge
0{,}98 \\[3mm]
&\Rightarrow \hspace*{5mm}
\frac{200 - Np}{\sqrt{Np(1-p)}} \ge \Phi^{-1}(0{,}98) =
2{,}06 \\[3mm]
&\Rightarrow \hspace*{5mm}
Np + 2{,}06\cdot \sqrt{Np(1-p)} - 200 \le 0
\end{align*}
\end{minipage}
\begin{minipage}[t]{0.4\textwidth}
\centering
\vspace*{-18mm}
\pause
\begin{align*}
u &:= \sqrt{N} \\
a &:= p = 0{,}03 \\
b &:= 2{,}06 \cdot \sqrt{p(1-p)} \approx 0{,}351 \\
c &:= -200
\end{align*}
\begin{gather*}
\rightarrow \hspace*{5mm} au^2 + bu + c \le 0 \\
\end{gather*}
\vspace*{-20mm}
\pause
\begin{align*}
\hspace{-17mm}\Rightarrow \hspace{3mm} u = \sqrt{N} \in \left[ -76, 87{,}7 \right] \\
\end{align*}
\vspace*{-25mm}
\pause\begin{align*}
&\Rightarrow \hspace{3mm} N \le 5776 \hspace{5mm} \cap \hspace{5mm} N \le 7691{,}29 \\
&\Rightarrow \hspace{3mm} N \le 5776
\end{align*}
\end{minipage}
\end{frame} \end{frame}
\end{document} \end{document}