From 475eead503ae1731a469244ed9063e908b73876c Mon Sep 17 00:00:00 2001 From: Luca Weidlich Date: Sat, 18 Jul 2026 07:50:02 +0200 Subject: [PATCH] kleinen Darstellungsfehler behiben --- content/Mathematischer_Hintergrund.tex | 4 ++-- galton.pdf | Bin 827966 -> 870705 bytes 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/content/Mathematischer_Hintergrund.tex b/content/Mathematischer_Hintergrund.tex index 1d82639..e626046 100644 --- a/content/Mathematischer_Hintergrund.tex +++ b/content/Mathematischer_Hintergrund.tex @@ -83,11 +83,11 @@ Eine weitere in diesem Kontext interessante Verteilung ist die Normalverteilung. \[ X \sim \mathcal{N}(\mu, \sigma ^2)\] Die Dichte der Normalverteilung wird auch ``Gauß’sche Glockenkurve'' genannt. \end{definition} -Für große $n$ wird die direkte Berechnung der Binomialverteilung aufgrund der Fakultäten sehr aufwendig. An dieser Stelle greift der zentrale Grenzwertsatz, der besagt, dass sich die Binomialverteilung für große $n$ der Normalverteilung annähert. Für große $n$ lässt sich damit die Wahrscheinlichkeit eines Ereignisses (aufgrund der Standardisierung) deutlich leichter bestimmen. Des Weiteren ist es mathematisch gesehen sehr schön zu sehen, wie eine relativ einfache diskrete Verteilung, wie die Binomialverteilung, mit einer mathematisch per se sehr anders aussehenden Verteilung zusammenhängt. Im Spezialfall der Binomialverteilung wird dies durch den Satz von Moivre-Laplace formalisiert \cite{nahrstedt_monte-carlo-methode_2015, statistik_verstehen_beweis_2019}. +Es ist mathematisch gesehen sehr schön zu sehen, wie eine relativ einfache diskrete Verteilung, wie die Binomialverteilung, mit einer mathematisch per se sehr anders aussehenden Verteilung zusammenhängt. Im Spezialfall der Binomialverteilung wird dies durch den Lokalen Grenzwertsatz von Moivre-Laplace formalisiert \cite{nahrstedt_monte-carlo-methode_2015, statistik_verstehen_beweis_2019}. \begin{satz}[Lokaler Grenzwertsatz von de Moivre-Laplace]{moivrelaplace}{nach \cite{georgii_stochastik_2015}} Sei $S_n \sim \mathcal{B}in_{n,p}$ die Anzahl der Erfolge bei $n$ unabhängigen Bernoulli-Versuchen mit Erfolgswahrscheinlichkeit $p \in (0, 1)$ und sei $q = 1-p$. Dann gilt bei einem beliebigen $c>0$ mit $z=\frac{k-np}{\sqrt{npq}}$, dass \[ - \lim_{n \to \infty} \mathop{\max}\limits_{k:|z|\leq c} \left|\frac{P(S_n=k)}{f_\mathcal{N(\mu, \sigma^2)}(k)}\right| = 1 + \lim_{n \to \infty} \mathop{\max}\limits_{k:|z|\leq c} \left|\frac{P(S_n=k)}{f_\mathcal{N}(\mu, \sigma^{2})}(k)\right| = 1 \] \end{satz} In der Literatur wird allgemein vom Satz von Moivre-Laplace gesprochen, wenn die Approximation der Binomial- durch die Normalverteilung gemeint ist \cite{statistik_verstehen_beweis_2019}. Mathematisch präzise muss jedoch zwischen dem globalen und dem lokalen Grenzwertsatz unterschieden werden \cite{georgii_stochastik_2015, kosenkova_stochastik_2025}. Wir beweisen im Folgenden den lokalen Grenzwertsatz, der das (lokale) Annähern des ``oberen Randes'' des Histogramms der Binomialverteilung an die Glockenkurve beschreibt \cite{georgii_stochastik_2015}. 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