From 2a8ba18a27c465975215f4d07f30c718371e3487 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Wed, 22 Apr 2026 11:16:39 +0200 Subject: [PATCH] [PS] Restructure, central limit theorem --- .../parts/05_limit-theorems/00_intro.tex | 7 ------- .../01_weak-law-of-large-numbers.tex | 0 .../02_strong-law-of-large-numbers.tex | 0 .../03_central-limit-theorem.tex | 19 ++++++++++++++++++ .../probability-and-statistics-cheatsheet.pdf | Bin 384803 -> 386713 bytes .../probability-and-statistics-cheatsheet.tex | 3 +++ 6 files changed, 22 insertions(+), 7 deletions(-) create mode 100644 semester4/ps/ps-jh/parts/05_limit-theorems/01_weak-law-of-large-numbers.tex create mode 100644 semester4/ps/ps-jh/parts/05_limit-theorems/02_strong-law-of-large-numbers.tex create mode 100644 semester4/ps/ps-jh/parts/05_limit-theorems/03_central-limit-theorem.tex 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 d2ced9d..5ba5ec2 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 @@ -3,10 +3,3 @@ {\scriptsize \shortterm In Zusammenhang mit Zufallsvariablen auch \bi{Sichprobenmittel} genannt. Realisierung wird \bi{empirischer Mittelwert} genannt} % TODO: Possibly add the remark from P288 (P6 in Slide Deck 6) on expected value - -\shorttheorem[Schwaches Ges. der grossen Zahlen] Sei $K = \{ 1, 2, \ldots \}$ und $\forall k \in K : \cX_k$ unabh. Z.V. mit $\E[\cX_k] = \mu$; $\V[\cX_k] = \sigma^2$: -\[ - \overline{\cX}_n = \frac{1}{n} S_n = \frac{1}{n} \sum_{k = 1}^{n} \cX_k -\] -Dann konvergiert $\overline{\cX}_n$ für $n \rightarrow \8$ in Wahrscheinlichkeit gegen $\mu = \E[\cX_k]$, -also $\forall \varepsilon > 0$ gilt $\P[|\overline{\cX}_n - \mu| > \varepsilon] \overset{n \rightarrow \8}{\longrightarrow} 0$ 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 new file mode 100644 index 0000000..e69de29 diff --git a/semester4/ps/ps-jh/parts/05_limit-theorems/02_strong-law-of-large-numbers.tex b/semester4/ps/ps-jh/parts/05_limit-theorems/02_strong-law-of-large-numbers.tex new file mode 100644 index 0000000..e69de29 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 new file mode 100644 index 0000000..ab93e25 --- /dev/null +++ b/semester4/ps/ps-jh/parts/05_limit-theorems/03_central-limit-theorem.tex @@ -0,0 +1,19 @@ +\subsection{Zentraler Grenzwertsatz} +\shortdefinition[Konvergenz in Verteilung] $(\cX_n)_{n \in \N}$, $\cX$ mit V.F.\\ +$(F_n)_{n \in \N}$, $F$. +$(\cX_n)_{n \in \N}$ \bi{konvergiert in V.} gegen $\cX$ +($\cX_n \overset{d}{\rightarrow} \cX$ für $n \rightarrow \8$), falls $\forall$ Stetigkeitsp. $x \in \R$ von $F$ gilt: +\[ + \limit{n}{\8} F_n(x) = \limit{n}{\8} \P[\cX_n \leq x] = \P[\cX \leq x] = F(x) +\] + +\shorttheorem[Zentraler Grenzwertsatz (\textbf{ZGS})]\\ +{\scriptsize i.i.d = independent and identically distributed (u.i.v in DE)} + +$\cX_k$ i.i.d mit $\E[\cX_k] = \mu$, $\V[\cX_k] = \sigma^2$. Für Partialsummen $S_n = \sum_{k = 1}^{n} \cX_k$ gilt $\forall x \in \R$ (mit $\Phi$ V.F. von Std.-Norm.-V): +\[ + \limit{n}{\8} \P \left[ \frac{S_n - n\mu}{\sigma \sqrt{n}} \leq x \right] = \Phi(x) +\] + +\shortremark $\E[S_n] = n \mu$, $\V[S_n] = n \sigma^2$; $S_n^* = \frac{S_n - \mu}{\sigma \sqrt{n}} \overset{\text{approx}}{\sim} \cN(0, 1)$ +für grosse $n$, mit $\overset{\text{approx}}{\sim}$ gespr. ``approx. gleichverteilt gemäss'' diff --git a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf index 6e2af896afd4bf4ff75db840445d745aa062c7b8..08618551acc3626d89c94384dbdcea46caacab12 100644 GIT binary patch delta 122459 zcmV(>K-j;dz89Iz7qCwd0XVZz5V!#WPO}LT)B%5hv{Io=@Fu)!8SuDm=iw6v!Kr)A zKMM|Q)2@PM+jl*wEdamad4>Pc;;83U2xKn3$JIA+=>JvuX6!lL4$wWO+u7_qrVVx` z1ThKpn#XYdL=}C$IKQ0F9xuHtGN(+b=`-NU?nrTd0*A{?zfT@J%U$Pq z@|u7CS_d5uAIK%yv;GqmBK@MCh5$c2an0Y%m&@x%UH0C*IRj+nj5)cjP?fOrl#$ju 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