From aa821d3261e30aefd12ca5522755f58b4ab32fc7 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Sat, 20 Jun 2026 09:04:33 +0200 Subject: [PATCH] [PS] fix: confidence intervals, limit theorem error --- .../01_distribution-function.tex | 1 + .../01_weak-law-of-large-numbers.tex | 1 + .../03_central-limit-theorem.tex | 4 ++-- .../08_confidence-intervals/00_regions.tex | 12 ++++++++---- .../08_confidence-intervals/01_interval.tex | 18 +++++++++--------- .../probability-and-statistics-cheatsheet.pdf | Bin 1120799 -> 1120185 bytes 6 files changed, 21 insertions(+), 15 deletions(-) diff --git a/semester4/ps/ps-jh/parts/01_random-variables/01_distribution-function.tex b/semester4/ps/ps-jh/parts/01_random-variables/01_distribution-function.tex index 2d49efe..f3f054c 100644 --- a/semester4/ps/ps-jh/parts/01_random-variables/01_distribution-function.tex +++ b/semester4/ps/ps-jh/parts/01_random-variables/01_distribution-function.tex @@ -1,5 +1,6 @@ % P23 \subsection{Verteilungsfunktion} +\label{sec:distribution-function} {\scriptsize Die untenstehende Funktion ist die Cumulative Distribution Function (cdf)} \shortdefinition $F_\cX : \R \rightarrow [0, 1]$, def: $\forall a \in \R \quad F_\cX(a) = \P[\cX \leq a]$ diff --git a/semester4/ps/ps-jh/parts/05_limit-theorems/01_weak-law-of-large-numbers.tex b/semester4/ps/ps-jh/parts/05_limit-theorems/01_weak-law-of-large-numbers.tex index 58a26a3..8626d0d 100644 --- a/semester4/ps/ps-jh/parts/05_limit-theorems/01_weak-law-of-large-numbers.tex +++ b/semester4/ps/ps-jh/parts/05_limit-theorems/01_weak-law-of-large-numbers.tex @@ -1,4 +1,5 @@ \subsection{Schwaches Gesetz der grossen Zahlen} +\label{sec:weak-law-of-large-numbers} \shorttheorem[Schwaches GGZ] Sei $K = \{ 1, 2, \ldots \}$ und $\forall k \in K : \cX_k$ unabh. Z.V. mit $\E[\cX_k] = \mu$; $\V[\cX_k] = \sigma^2$: \[ \overline{\cX}_n = \frac{1}{n} S_n = \frac{1}{n} \sum_{k = 1}^{n} \cX_k diff --git a/semester4/ps/ps-jh/parts/05_limit-theorems/03_central-limit-theorem.tex b/semester4/ps/ps-jh/parts/05_limit-theorems/03_central-limit-theorem.tex index c1caac9..7d8e6f1 100644 --- a/semester4/ps/ps-jh/parts/05_limit-theorems/03_central-limit-theorem.tex +++ b/semester4/ps/ps-jh/parts/05_limit-theorems/03_central-limit-theorem.tex @@ -19,10 +19,10 @@ $\cX_k$ i.i.d mit $\E[\cX_k] = \mu$, $\V[\cX_k] = \sigma^2$. Für Partialsummen für grosse $n$, mit $\overset{\text{approx}}{\sim}$ gespr. ``approximativ verteilt gemäss''. Also ist für $\E[S_n^*] = 0$ und $\V[S_n^*] = 1$. -Für $S_n$ also: $S_n \! \overset{\text{approx}}{\sim} \! \cN(n\mu, n\sigma^2)$, +Also: $S_n \! \overset{\text{approx}}{\sim} \! \cN(n\mu, n\sigma^2)$, bzw. $\overline{\cX}_n \! \overset{\text{approx}}{\sim} \! \cN \left( \mu, \frac{1}{n} \sigma^2 \right)$ -\shortremark Für $S_n \sim \text{Bin}(n, p)$ ist $S_n \overset{\text{approx}}{\sim} \cN(np, np(1 - p))$ und +\shortremark Für $S_n \sim \text{Bin}(n, p)$ ist $S_n^* \overset{\text{approx}}{\sim} \cN(np, np(1 - p))$ und $\P[a < S_n \leq b] \approx \Phi \left( \frac{b + \frac{1}{2} - np}{\sqrt{np(1 - p)}} \right) - \Phi \left( \frac{a + \frac{1}{2} - np}{\sqrt{np(1 - p)}} \right)$ \shortremark Für $\P[S_n \leq y]$: ZGS verwenden mit $\displaystyle x = \frac{y - n\mu}{\sigma \sqrt{n}}$, diff --git a/semester4/ps/ps-jh/parts/08_confidence-intervals/00_regions.tex b/semester4/ps/ps-jh/parts/08_confidence-intervals/00_regions.tex index 14b6cc1..61197f2 100644 --- a/semester4/ps/ps-jh/parts/08_confidence-intervals/00_regions.tex +++ b/semester4/ps/ps-jh/parts/08_confidence-intervals/00_regions.tex @@ -1,8 +1,12 @@ -\shortdefinition[Konfidenzbereich] ist für $\vartheta$ zu Daten $x_i$ eine Menge $C(x_1, \ldots, x_n) \subseteq \Theta$. -Mit genrierenden Z.V. ist $\tilde{C}(\cX_1, \ldots \cX_n)$ eine \bi{zufällige Teilmenge von} $\Theta$. +\shortdefinition[Konfidenzbereich] $C(x_1, \ldots, x_n) \subseteq \Theta$. +Mit genrierenden Z.V. ist $\tilde{C} := C(\cX_1, \ldots \cX_n)$ eine \bi{zufällige Teilmenge von} $\Theta$, +{\scriptsize (\textit{Realisierung}) $\tilde{C}(\omega) = C(\cX_1(\omega), \ldots, \cX_n(\omega))$} -\bi{Realisierung} $\tilde{C}(\omega) = C(\cX_1(\omega), \ldots, \cX_n(\omega))$. Für $\alpha \in [0, 1]$ ist $C$ \bi{Konfidenzbereich zum Niveau} $1 - \alpha$, falls \[ - \P_\vartheta[C(\cX_1, \ldots, \cX_n) \ni \vartheta] \geq 1 - \alpha \quad \forall \vartheta \in \Theta + \P_\vartheta[C(\cX_1, \ldots, \cX_n) \ni \vartheta] \geq 1 - \alpha \quad \forall \vartheta \in \Theta \] +{\scriptsize Man erwischt also mit grosser W. den Param in jedem Modell} + +\shortexample $C(\cX_1, \ldots) = [\overline{X}_n - c, \overline{X}_n + c]$, mit $c$ zu bestimmen. + diff --git a/semester4/ps/ps-jh/parts/08_confidence-intervals/01_interval.tex b/semester4/ps/ps-jh/parts/08_confidence-intervals/01_interval.tex index e0f171c..985428a 100644 --- a/semester4/ps/ps-jh/parts/08_confidence-intervals/01_interval.tex +++ b/semester4/ps/ps-jh/parts/08_confidence-intervals/01_interval.tex @@ -1,31 +1,31 @@ \shortdefinition[Konfidenzintervall (K.I.)] mit Niveau $1 - \alpha$ ist Zufallsintervall $I = [A, B]$ s.d. $\forall \vartheta \in \Theta$ gilt: \[ - \P_\vartheta[[A, B] \ni \vartheta] = \P_\vartheta[A \leq \vartheta \leq B] \geq 1 - \alpha + \P_\vartheta[[A, B] \ni \vartheta] = \P_\vartheta[A \leq \vartheta \leq B] \geq 1 - \alpha \] mit $A, B$ Z.V. wie $A = a(\cX_1, \ldots, \cX_n)$, mit $a, b: \R^n \rightarrow \R$. -Dabei ist $\vartheta$ deterministisch +Dabei ist $\vartheta$ deterministisch. (\hl{Berechnung}: s. \ref{sec:distribution-function}) \shortremark[Approx. K.I.] Nur wenn genaue Verteilungsaussagen zur Verfügung stehen, sind exakte K.I. möglich, sonst mittels zentralem Grenzwertsatz approx. berechnen ($z = z_{1 - \frac{\alpha}{2}}$) \[ - P_\vartheta[|S_n^*| \leq z] \approx 1 - \alpha + P_\vartheta[|S_n^*| \leq z] \approx 1 - \alpha \] -Auflösen von +Auflösen von untenstehendem nach $\vartheta$ ($S_n$ wie in \ref{sec:weak-law-of-large-numbers}) \[ - |S_n - n \vartheta| \leq z \sqrt{n \vartheta(1 - \vartheta)} \text{ bzw. } (S_n - n \vartheta)^2 \leq z^2 n \vartheta(1 - \vartheta) + |S_n - n \vartheta| \leq z \sqrt{n \vartheta(1 - \vartheta)} \text{ bzw. } (S_n - n \vartheta)^2 \leq z^2 n \vartheta(1 - \vartheta) \] -$\vartheta$ ist kompliziert, ersetzen der Ugl. durch Gl. (quad. Gl.) +Ist kompliziert, ersetzen der Ugl. durch Gl. (quad. Gl.) \[ - \hat{\vartheta}_\pm = \frac{2 S_n + z^2 \pm \sqrt{(2 S_n + z^2)^2 - 4S_n^2(1 + \frac{z^2}{n})}}{2n(1 + \frac{z^2}{n})} + \hat{\vartheta}_\pm = \frac{2 S_n + z^2 \pm \sqrt{(2 S_n + z^2)^2 - 4S_n^2(1 + \frac{z^2}{n})}}{2n(1 + \frac{z^2}{n})} \] Konfidenzintervall ist $[\hat{\vartheta}_-, \hat{\vartheta}_+]$. \shade{gray}{Alternative Methoden} (Annahme $\vartheta(1 - \vartheta) \approx \frac{1}{4}$): \[ - \left[\overline{S}_n - \frac{z}{2 \sqrt{n}}, \overline{S}_n + \frac{z}{2 \sqrt{n}}\right] + \left[\overline{S}_n - \frac{z}{2 \sqrt{n}}, \overline{S}_n + \frac{z}{2 \sqrt{n}}\right] \] (Ersetzen von $\vartheta$ durch $\overline{S}_n$): \[ - \left[ \overline{S}_n - \frac{z}{2 \sqrt{n}} \sqrt{\overline{S}_n (1 - \overline{S}_n)}, \overline{S}_n + \frac{z}{2 \sqrt{n}} \sqrt{\overline{S}_n (1 - \overline{S}_n)} \right] + \left[ \overline{S}_n - \frac{z}{2 \sqrt{n}} \sqrt{\overline{S}_n (1 - \overline{S}_n)}, \overline{S}_n + \frac{z}{2 \sqrt{n}} \sqrt{\overline{S}_n (1 - \overline{S}_n)} \right] \] Alle Methoden liefern unterschiedliche Resultate diff --git a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf index 63207d32f8df3e734307eecf19912daa51bdb484..7807ee6841ac0b48a975b4d7d4e7d3d972bc36a4 100644 GIT binary patch delta 67480 zcmV)4K+3TfrVQp{2;y$p4~h5GzZcir6Uyd)Kat$b zmp7M1h1xA7!KN@p0jJl&Gea5UGfIr&T;W18X#k_E4(bP|vjZ=P1Y;ZncgD3aCe0uMXkGT{B-*_X;;OWZ&w+&-%V7O?M96UW6JRj;t z9vySW*zoD(W_ENer5yeneyHKoBd3P9lhaqTqo1Gtd0+%@q*l;&c+jNB9v%rd{Nf}& z`Q*{bqo>c};}0Kwf0r(P_dh2mkNz6}c=pAA{RO^5gZCaiz4yyK4(+i4Uv!PR#way7 zy}I}OXEr#4-~K#cjr4BtCfs>7aLhK&Vm~eiC-=U+$I^J)&y_ZhSz}@RTQnI+=T$~I z#a*Ld@W((p=Mw3{D7vpln)B?*W5sx*lo@De8V5~;XCF;gJekc-;rk1q z49CAOyjL|*u?W=t^M#Prdj+IdY=6_e3`E5;t|e8{gJm1VD`Ko{z^V74L-@C zLqZQfPq&zXYP{nz@R1aa5o!Sc+C&UaOsVn>H*TP z_!E;nk&E$ce*m_T!a?5_w=qr*Qy4aEf5~6w8m`6QSTq8f4NbfyXdoL8YY3*+*m#o& zt3(2Ocwm#|^-qacJYP`W6fgGDt~80m6UB?WZ}^D`MKt%k9bL&yz7b9jveAXPj*r`% 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