From 865c004c7540d72fde0d2fcf42d295b326ea456f Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Mon, 30 Mar 2026 11:54:00 +0200 Subject: [PATCH] [PS] Some fixes --- .../05_cont-distributions/03_normal.tex | 6 ++++-- .../ps-jh/parts/03_expected-value/02_cont.tex | 2 +- .../probability-and-statistics-cheatsheet.pdf | Bin 344966 -> 344420 bytes 3 files changed, 5 insertions(+), 3 deletions(-) diff --git a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/03_normal.tex b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/03_normal.tex index f9c4dc8..4969e1c 100644 --- a/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/03_normal.tex +++ b/semester4/ps/ps-jh/parts/02_discrete-continuous-rv/05_cont-distributions/03_normal.tex @@ -3,13 +3,15 @@ mit $\sigma$ Standardabweichung. Auch: Gauss'sche Verteilung \shortdefinition[Standardnormalverteilung] $\cX \sim \cN(0, 1)$:\\ -$f_\cX = \varphi$ und $\F_\cX = \Phi = \int_{-\8}^{x} \varphi(t) \dx t = \frac{1}{\sqrt{2\phi}} \int_{-\8}^{x} e^\frac{-t^2}{2} \dx t$ +$\Phi = \int_{-\8}^{x} \varphi(t) \dx t = \frac{1}{\sqrt{2\pi}} \int_{-\8}^{x} e^\frac{-t^2}{2} \dx t$ mit $\varphi = f_\cX$.\\ +\bi{Es gilt}: $\Phi(-t) = 1 - \Phi(t)$ -\shorttheorem $cX \sim \cN(\mu, \sigma^2)$, dann $\frac{\cX - \mu}{\sigma} \sim \cN(0, 1)$, also: +\shorttheorem $\cX \sim \cN(\mu, \sigma^2)$, dann $\frac{\cX - \mu}{\sigma} \sim \cN(0, 1)$, also: \[ F_\cX(x) = \P[\cX \leq x] = \P\left[ \frac{\cX - \mu}{\sigma} \leq \frac{x - \mu}{\sigma} \right] = \Phi \left( \frac{x - \mu}{\sigma} \right) \] +\newpage \shortexample für Phänomene modellierbar mit Normalverteilung: \begin{itemize} \item Streuung von Messwerten um Mittelwert diff --git a/semester4/ps/ps-jh/parts/03_expected-value/02_cont.tex b/semester4/ps/ps-jh/parts/03_expected-value/02_cont.tex index ce2fbf6..9b68fd5 100644 --- a/semester4/ps/ps-jh/parts/03_expected-value/02_cont.tex +++ b/semester4/ps/ps-jh/parts/03_expected-value/02_cont.tex @@ -9,7 +9,7 @@ \begin{itemize} \item $\cX \sim \cU([a, b])$, $a < b$: $\E[\cX] = \frac{a + b}{2}$ \item $\cX \sim \text{Exp}(\lambda)$, $\lambda > 0$: $\E[\cX] = \frac{1}{\lambda}$ - \item $\cX \sim \cU(\mu, \sigma^2)$, $z = x - \mu$, $\dx z = \dx x$: $\E[\cX] = \mu$ + \item $\cX \sim \cN(\mu, \sigma^2)$: $\E[\cX] = \mu$ \item $\cX \sim \text{Cauchy}(x_0, \gamma)$: Existiert nicht (Int. $\8$)\\ $\E[\cX_+] = \E[\cX_-] = \8$, Median: $0$ \end{itemize} diff --git a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf index 1c64b96203e8a219e0b8a702403ae03fef4eb815..425f9f2b3c1829f8d57318134ed1031d60d74696 100644 GIT binary patch delta 63601 zcmV)8K*qm@hZW?46|e~sf93_jA=UJ`osfqIw#x(cfA%r_$>e1FaK8TNUSb7P(GJ*^ 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