From c5ddc88e121c8c10ff40599f598d36122cf93cd3 Mon Sep 17 00:00:00 2001 From: Luca Weidlich Date: Mon, 20 Jul 2026 08:22:30 +0200 Subject: [PATCH] =?UTF-8?q?letzte=20=C3=84nderungen=20wegen=20LGS?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- content/Weiterfuehrende_Ideen.tex | 5 ++++- galton.pdf | Bin 870705 -> 871216 bytes 2 files changed, 4 insertions(+), 1 deletion(-) diff --git a/content/Weiterfuehrende_Ideen.tex b/content/Weiterfuehrende_Ideen.tex index c4f990f..09bf978 100644 --- a/content/Weiterfuehrende_Ideen.tex +++ b/content/Weiterfuehrende_Ideen.tex @@ -1,4 +1,7 @@ -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, sowie praktische Erweiterungsmöglichkeiten, die weit über das Standardmodell hinausgehen. +Das klassische Galtonbrett ist ein elegantes mechanisches Modell zur Veranschaulichung des lokalen und zentralen Grenzwertsatzes. 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, sowie praktische Erweiterungsmöglichkeiten, die weit über das Standardmodell hinausgehen. +\paragraph{Zentraler Grenzwertsatz} +In dieser Arbeit wurde der Fokus auf den lokalen Grenzwertsatz von Movire-Laplace gelegt. Es ist ebenso möglich dieses Resultat weiter zum zentralen Grenzwertsatz von Movire-Laplace zu erweitern. Hiermit lässt sich zeigen, dass sich die Binomialverteilung mithilfe der Normalverteilung approximieren lässt \cite{kosenkova_stochastik_2025}. + \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. 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