From 5c8ff7b3a7d5f9c8829a1119b8374e5ceb36b4c0 Mon Sep 17 00:00:00 2001 From: Luca Weidlich Date: Sun, 12 Jul 2026 16:04:59 +0200 Subject: [PATCH] Weiteres Dings Kapitel und kleinigkeiten --- bibliography.bib | 28 +++++++++++++++++++++++++ content/Mathematischer_Hintergrund.tex | 2 +- content/Weiterfuehrende_Ideen.tex | 13 ++++++++---- galton.pdf | Bin 493249 -> 479415 bytes 4 files changed, 38 insertions(+), 5 deletions(-) diff --git a/bibliography.bib b/bibliography.bib index 7f6d858..8a1fdce 100644 --- a/bibliography.bib +++ b/bibliography.bib @@ -133,3 +133,31 @@ address = {Humboldt-Universität zu Berlin}, url = {https://www2.informatik.hu-berlin.de/~koessler/Stochastik/Stochastik2006/FolienStochastik_4.pdf} } + + +@article{schreiter_echt_2022, + title = {Echt schräg: Galton revisited}, + url = {https://www.friedrich-verlag.de/friedrich-plus/sekundarstufe/mathematik/stochastik/echt-schrag-galton-revisited-12235}, + number = {232}, + publisher = {Friedrich Verlag {GmbH}}, + author = {Schreiter, Saskia and Vogel, Markus and Eichler, Andreas}, + urldate = {2026-07-12}, + date = {2022}, +} + + +@incollection{posedel_martingale_2025, + location = {Berlin, Heidelberg}, + title = {Martingale Central Limit Theorem}, + isbn = {978-3-662-69359-9}, + url = {https://doi.org/10.1007/978-3-662-69359-9_345}, + doi = {10.1007/978-3-662-69359-9_345}, + abstract = {The martingale central limit theorem ({MCLT}) links the notions of martingales and the Lindeberg-Feller classical central limit theorem ({CLT}) for independent summands.}, + pages = {1418--1421}, + booktitle = {International Encyclopedia of Statistical Science}, + publisher = {Springer Berlin Heidelberg}, + author = {Posedel, Petra}, + editor = {Lovric, Miodrag}, + date = {2025}, +} + diff --git a/content/Mathematischer_Hintergrund.tex b/content/Mathematischer_Hintergrund.tex index f6bcbd1..3309762 100644 --- a/content/Mathematischer_Hintergrund.tex +++ b/content/Mathematischer_Hintergrund.tex @@ -88,7 +88,7 @@ Für große $n$ wird die direkte Berechnung der Binomialverteilung aufgrund der \[ P(S_n = k) \sim \mathcal{N}(\mu,\sigma ^2) \] - wobei $\mathbb{E}(X)=\mu$, $Var(X)=\sigma ^2$ und $\frac{k-np}{\sqrt{npq}}$ begrenzt ist. + wobei $\mathbb{E}(X)=\mu$, $Var(X)=\sigma ^2 < \infty$ und $\frac{k-np}{\sqrt{npq}}$ begrenzt ist. \end{satz} \begin{proof} Der Beweis folgt zum großen Teil dem Beweis von \cite{statistik_verstehen_beweis_2019} und wurde von den Autoren weiter konkretisiert.\\ diff --git a/content/Weiterfuehrende_Ideen.tex b/content/Weiterfuehrende_Ideen.tex index ba20b6c..2802f66 100644 --- a/content/Weiterfuehrende_Ideen.tex +++ b/content/Weiterfuehrende_Ideen.tex @@ -1,4 +1,9 @@ -Da Sie ja umfangreich recherchiert haben, werden Sie deutlich mehr Material haben, als Sie in diesem Dokument unterbringen können. Hier können Sie Ihre weiterführenden Ideen kurz darstellen. Vergessen Sie nicht, auch hier die Quellen anzugeben! -% Vorraussetzungen entspannter betrachten: -% Martingale central limit theorem + Mixing Condition -> Normalverteilung. -% Gegenbeispiel Fat-Tails -> was geht kaputt? \ No newline at end of file +Das klassische Galtonbrett ist ein elegantes mechanisches Modell zur Veranschaulichung des zentralen Grenzwertsatzes (ZGS). Die Standardannahme geht dabei von unabhängigen, identisch verteilten Zufallsvariablen aus: jede Kugel trifft an jedem Stift mit einer Wahrscheinlichkeit von $p = 0.5$ die Entscheidung, nach links oder rechts zu fallen ohne, dass diese sich gegenseitig beeinflussen. Im Rahmen tiefergehender Recherchen zeigen sich jedoch spannende theoretische wie praktische Erweiterungsmöglichkeiten, die weit über das Standardmodell hinausgehen. +\paragraph{Unabhängigkeit} +In der Realität ist die strikte Unabhängigkeit aufeinanderfolgender Schritte oft eine Idealisierung. Physikalisch gesehen könnte eine Kugel den Stift in leichte Schwingungen versetzen oder Drall mitnehmen, was die Wahrscheinlichkeit für den nächsten Schritt beeinflusst. Mathematisch lässt sich diese Voraussetzung jedoch lockern, indem statt Zufallsvariablen Martingale betrachtet werden. Unter bestimmten Voraussetzungen greif hier dann der Martingal-Zentralgrenzwertsatz, wodurch wieder die Normalverteilung angenähert wird \cite{posedel_martingale_2025}. + +\paragraph{Identisch Verteilt} +Das klassische Brett nutzt auf jeder Ebene die identische Wahrscheinlichkeit $p = 0.5$. Da auch dies nicht unbedingt immer Gewehrleitbar ist, lässt sich diese Bedingung ebenso aufweichen. Unter der Bedingung, dass keine einzelne Reihe die Gesamtvarianz dominiert (Lindeberg-Bedingung), führt der verallgemeinerte zentrale Grenzwertsatz dennoch zu einer Normalverteilung. Die Glockenkurve bleibt erhalten, auch wenn die Elementarschritte auf dem Weg nach unten variieren \cite{posedel_martingale_2025}. + +\paragraph{Erweiterung der Mechanik in der Schule: Das schiefe Galtonbrett} +Eine physisch leicht umzusetzende Variation ist das Neigen des gesamten Brettes. Durch die Schräglage wird die Wahrscheinlichkeit für einen Fall nach rechts oder links asymmetrisch, sodass $p \neq 0.5$ gilt. Die Pfade der Kugeln folgen nun einer asymmetrischen Binomialverteilung. Für eine große Anzahl an Stiftreihen $n$ nähert sich die Form (nach dem Satz von Moivre-Laplace) zwar immer noch einer Normalverteilung an, jedoch verschieben sich der Erwartungswert auf $\mu = np$ und die Varianz auf $\sigma^2 = np(1-p)$. 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