From 7ca8453e805945d2f339f276b198d1e4e2e38677 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Wed, 19 Aug 2026 10:53:17 +0200 Subject: [PATCH] [PS] Layout fixes, new remarks --- .../parts/05_limit-theorems/00_intro.tex | 2 +- .../parts/06_estimators/03_properties.tex | 2 ++ .../ps/ps-jh/parts/07_tests/02_examples.tex | 4 +++- .../parts/10_tips-and-tricks/01_various.tex | 2 +- .../probability-and-statistics-cheatsheet.pdf | Bin 1134841 -> 1135521 bytes 5 files changed, 7 insertions(+), 3 deletions(-) diff --git a/semester4/ps/ps-jh/parts/05_limit-theorems/00_intro.tex b/semester4/ps/ps-jh/parts/05_limit-theorems/00_intro.tex index 922b6c6..16f352c 100644 --- a/semester4/ps/ps-jh/parts/05_limit-theorems/00_intro.tex +++ b/semester4/ps/ps-jh/parts/05_limit-theorems/00_intro.tex @@ -4,4 +4,4 @@ \shortremark \hl{Vereinfacht}: Beide besagen, dass $\limit{n}{\8} \overline{\cX}_n \approx \E[\cX_k]$ -% TODO: Possibly add the remark from P288 (P6 in Slide Deck 6) on expected value +\shortremark Erwartungswert und Varianz mit lin, etc berechnen diff --git a/semester4/ps/ps-jh/parts/06_estimators/03_properties.tex b/semester4/ps/ps-jh/parts/06_estimators/03_properties.tex index 5776b5c..0193521 100644 --- a/semester4/ps/ps-jh/parts/06_estimators/03_properties.tex +++ b/semester4/ps/ps-jh/parts/06_estimators/03_properties.tex @@ -16,6 +16,8 @@ Dabei $T \sim \cN(n \E_\vartheta[\cY_k], n \V_\vartheta[\cY_k])$ unter $\P_\vart \shortremark $\cX_k \sim \cN(0, 1)$ i.i.d: $\displaystyle \left( \sum_{k = 1}^{m} \cX_k^2 \right) \sim \chi^2_m$. $\chi^2_2 = \text{Exp}(\frac{1}{2})$ +\shortremark Allgemein: $\cX_k \sim \cN(\mu, \sigma^2)$, dann $\frac{n - 1}{\sigma^2} S^2 \sim \chi_{n - 1}^2$, mit $S$ eine Summe wie oberhalb + \shortdefinition[Studentsche $t$-Verteilung] $\cX \sim t_m$ falls Dichte \[ f_\cX(x) = \frac{\Gamma\left( \frac{m + 1}{2} \right)}{\sqrt{m \pi} \Gamma \left( \frac{m}{2} \right)} \left( 1 + \frac{x^2}{m} \right)^{-\frac{m + 1}{2}} diff --git a/semester4/ps/ps-jh/parts/07_tests/02_examples.tex b/semester4/ps/ps-jh/parts/07_tests/02_examples.tex index 3c0fa8e..fe4a873 100644 --- a/semester4/ps/ps-jh/parts/07_tests/02_examples.tex +++ b/semester4/ps/ps-jh/parts/07_tests/02_examples.tex @@ -14,9 +14,11 @@ \item Testentscheid: Basierend auf Resultat von vorherigem und $K$ entscheiden. \end{enumerate} } -\bi{Verwerfungsbereich} $K_l = (c_>, \8)$ (links), $K_r = (-\8, c_<)$ (rechts), $K_b = (-\8, -c_{\neq}) \cup (c_{\neq}, \8)$ (beidseitig) +\bi{Approximative Teststatistik} Normalisierung unter $H_0$, also $T = \frac{x_n - \E[S_n]}{\V[S_n]}$, mit $x_n$ Realisierung, $S_n$ wie immer \newpage +\bi{Verwerfungsbereich} $K_l = (c_>, \8)$ (links), $K_r = (-\8, c_<)$ (rechts), $K_b = (-\8, -c_{\neq}) \cup (c_{\neq}, \8)$ (beidseitig) + \shortexample[Gauss-Test] Voraussetz.: \bi{(1)} $\cX_i$ i.i.d. $\sim \cN(\vartheta, \sigma^2)$ unter $\P_\vartheta$ \bi{(2)} $\sigma^2$ bekannt. diff --git a/semester4/ps/ps-jh/parts/10_tips-and-tricks/01_various.tex b/semester4/ps/ps-jh/parts/10_tips-and-tricks/01_various.tex index d4f2357..af8a0f4 100644 --- a/semester4/ps/ps-jh/parts/10_tips-and-tricks/01_various.tex +++ b/semester4/ps/ps-jh/parts/10_tips-and-tricks/01_various.tex @@ -1,4 +1,4 @@ -\rmvspace +\newpage \subsection{Verschiedene Funktionen} \shortremark[Ungleichungen] Zur Erinnerung: Richtung wechselt bei Muliplikation mit (Division durch) negative Zahl. diff --git a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf index f2812334bafbd19cb5288adc6f25957e7cb0938b..c87a45ac31462565e88c1a718fe186a99533734f 100644 GIT binary patch delta 116855 zcmV(#K;*yqp+%vmMX(kUe>4P#RP#qiFz(IS&Gn`Nfn`T@&Q!3zcHO8TT&N11ToOtZ zfTf9Hu^pMcUQmiCx4bToud7#t<_tHTP_5wKjEUL1S?uZeliJ2mnM(>gqY>}$ITMCx zK7;MjTFf`BbLvX_|HEP5xAINmly9><>@~^`|NXkz~?0?i2H4QU0iLdZfPdH zeqY=7Pa-3}o_}8TTWpS9hf9mE@Jl_^^i7JLE+1(bF@VMk*b|-Aq=>}vBsg<;;?{tL 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