From b815a88361c47bc83a169a9db0aef0fee6ba725c Mon Sep 17 00:00:00 2001 From: Andreas Tsouchlos Date: Wed, 17 Dec 2025 15:16:56 +0100 Subject: [PATCH] Replace Z with X for the standart normal distribution --- src/2025-12-19/presentation.tex | 59 ++++++++++++++++++--------------- 1 file changed, 32 insertions(+), 27 deletions(-) diff --git a/src/2025-12-19/presentation.tex b/src/2025-12-19/presentation.tex index 9a7d039..478c8b8 100644 --- a/src/2025-12-19/presentation.tex +++ b/src/2025-12-19/presentation.tex @@ -168,7 +168,7 @@ \begin{frame} \frametitle{Zusammenfassung} - \begin{columns}[c] + \begin{columns}[t] \column{\kitthreecolumns} \centering \begin{greenblock}{Verteilungsfunktion (kontinuierlich)} @@ -463,7 +463,6 @@ \end{columns} \end{frame} -% TODO: Are Z/z notation used in the lecture? \begin{frame} \frametitle{Rechnen mithilfe der Standardnormalverteilung} @@ -476,16 +475,16 @@ \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) + X \sim \mathcal{N} (0,1) \\[4mm] + \Phi(x) := F_X(x) = P(X \le x) \\ + \Phi(-x) = 1 - \Phi(x) \end{gather*} \end{minipage}% \begin{minipage}{0.48\textwidth} \centering \begin{tabular}{|c|c||c|c||c|c|} \hline - $z$ & $\Phi(z)$ & $z$ & $\Phi(z)$ & $z$ & $\Phi(z)$ \\ + $x$ & $\Phi(x)$ & $x$ & $\Phi(x)$ & $x$ & $\Phi(x)$ \\ \hline \hline 0{,}00 & 0{,}500000 & 0{,}10 & 0{,}539828 & 0{,}20 & 0{,}579260 \\ @@ -512,7 +511,7 @@ \end{gather*} \end{itemize} - \vspace*{5mm} + \vspace*{3mm} \pause \begin{lightgrayhighlightbox} @@ -521,13 +520,13 @@ 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 + = P\big(\underbrace{\widetilde{X}}_{\sim + \mathcal{N}(0,1)} \le 0{,}24\big) + = \Phi\left(0{,}24\right) = 0{,}594835 \end{gather*} \end{lightgrayhighlightbox} \end{frame} -% TODO: Are Z/z notation used in the lecture? \begin{frame} \frametitle{Zusammenfassung} @@ -539,9 +538,9 @@ \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) + 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} \column{\kitthreecolumns} @@ -562,14 +561,20 @@ % \cdots \begin{tabular}{|c|c||c|c||c|c||c|c|} \hline - $z$ & $\Phi(z)$ & $z$ & $\Phi(z)$ & $z$ & $\Phi(z)$ & $z$ & $\Phi(z)$ \\ + $x$ & $\Phi(x)$ & $x$ & $\Phi(x)$ & $x$ & $\Phi(x)$ & $x$ + & $\Phi(x)$ \\ \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 \\ + 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 @@ -631,8 +636,8 @@ P(E_\text{a}) &= P \Big( \big( X < S(1-\delta) \big) \cup \big( X > S(1 + \delta) \big) \Big) \\ &= P(X < S(1 - \delta)) + P(X > S(1 + \delta)) \\[2mm] - &= P\left(Z < \frac{S(1 - \delta) - \mu}{\sigma}\right) - + P\left(Z > \frac{S(1 + \delta) - \mu}{\sigma}\right) \\[2mm] + &\overset{\widetilde{X} := \frac{X - \mu}{\sigma} }{=\joinrel=\joinrel=\joinrel=} P\left(\widetilde{X} < \frac{S(1 - \delta) - \mu}{\sigma}\right) + + P\left(\widetilde{X} > \frac{S(1 + \delta) - \mu}{\sigma}\right) \\[2mm] &\approx \Phi(-2{,}86) + \left(1 - \Phi(2{,}86)\right) \\ &= 2 - 2\Phi(2{,}86) \approx 0{,}424\text{\%} \end{align*} @@ -687,10 +692,10 @@ \pause\column{\kitthreecolumns} \centering \begin{align*} - P(E_\text{b}) &\overset{\text{a)}}{=} P\left(Z < \frac{S(1 - \delta) - \mu}{\sigma'}\right) - + P\left(Z > \frac{S(1 + \delta) - \mu}{\sigma'}\right) \\[2mm] - &= P\left(Z < -\frac{0{,}2}{\sigma'}\right) - + P\left(Z > \frac{0{,}2}{\sigma'}\right) \\[2mm] + P(E_\text{b}) &\overset{\text{a)}}{=} P\left(\widetilde{X} < \frac{S(1 - \delta) - \mu}{\sigma'}\right) + + P\left(\widetilde{X} > \frac{S(1 + \delta) - \mu}{\sigma'}\right) \\[2mm] + &= P\left(\widetilde{X} < -\frac{0{,}2}{\sigma'}\right) + + P\left(\widetilde{X} > \frac{0{,}2}{\sigma'}\right) \\[2mm] &= \Phi\left(-\frac{0{,}2}{\sigma'}\right) + \left(1 - \Phi\left(\frac{0{,}2}{\sigma'} \right)\right) \\[2mm] &= 2 - 2 \Phi\left(\frac{0{,}2}{\sigma'} \right) @@ -708,8 +713,8 @@ Mittelwert $\mu$ auf $5{,}1$ Volt ($\sigma$ ist $0{,}07$ Volt). Wie groß ist jetzt der Prozentsatz, der aussortiert wird? \pause \begin{align*} - P(E_\text{c}) &\overset{\text{a)}}{=} P\left(Z < \frac{S(1 - \delta) - \mu}{\sigma}\right) - + P\left(Z > \frac{S(1 + \delta) - \mu}{\sigma}\right) \\[2mm] + P(E_\text{c}) &\overset{\text{a)}}{=} P\left(\widetilde{X} < \frac{S(1 - \delta) - \mu}{\sigma}\right) + + P\left(\widetilde{X} > \frac{S(1 + \delta) - \mu}{\sigma}\right) \\[2mm] &\approx \Phi(-4{,}29) + (1 - \Phi(1{,}43)) \\ & = 2 - \Phi(4{,}29) - \Phi(1{,}43) \approx 7{,}78 \text{\%} \end{align*}