From ea776b8f0f436172e1d0b8464051c0fd2a59c8a1 Mon Sep 17 00:00:00 2001 From: Elias Fierke Date: Sun, 5 Jul 2026 11:58:49 +0200 Subject: [PATCH] fix: typos und einheitliche Schreibweise 'Galtonbrett' --- content/Kernideen.tex | 4 ++-- content/Mathematischer_Hintergrund.tex | 26 ++++++++++++------------- galton.pdf | Bin 487212 -> 453963 bytes 3 files changed, 15 insertions(+), 15 deletions(-) diff --git a/content/Kernideen.tex b/content/Kernideen.tex index 219bc37..791d7e1 100644 --- a/content/Kernideen.tex +++ b/content/Kernideen.tex @@ -1,4 +1,4 @@ -In dieser Arbeit widmen wir uns der mathematischen Formalisierung eines faszinierenden Phänomens: Wie verteilen sich zufällige Ereignisse, wenn sie sehr oft wiederholt werden? Als anschauliches Modell dient uns hierfür das Galton-Brett, bei dem Kugeln an Hindernissen zufällig nach links oder rechts fallen. +In dieser Arbeit widmen wir uns der mathematischen Formalisierung eines faszinierenden Phänomens: Wie verteilen sich zufällige Ereignisse, wenn sie sehr oft wiederholt werden? Als anschauliches Modell dient uns hierfür das Galtonbrett, bei dem Kugeln an Hindernissen zufällig nach links oder rechts fallen. %Informell lautet unsere Fragestellung: Lässt sich das exakte, aber für große Zahlen unhandliche Zählverfahren der Kugelwege (Kombinatorik) durch eine einfachere, kontinuierliche Kurve beschreiben? @@ -8,4 +8,4 @@ Den größeren Kontext bildet die Verknüpfung grundlegender mathematischer Werk Das Herzstück der Arbeit und die formale Antwort auf unsere Fragestellung bildet der \emph{Zentraler Grenzwertsatz von de Moivre-Laplace}. Er besagt, dass sich die diskrete Binomialverteilung für große $n$ der stetigen Normalverteilung (der bekannten Gaußschen Glockenkurve) annähert. -Damit wird mathematisch präzise gezeigt, wie ein System aus einfachen, binären Entscheidungen (links oder rechts) bei ausreichend vielen Wiederholungen unweigerlich in ein universelles, kontinuierliches Verteilungsmodell mündet. \ No newline at end of file +Damit wird mathematisch präzise gezeigt, wie ein System aus einfachen, binären Entscheidungen (links oder rechts) bei ausreichend vielen Wiederholungen unweigerlich in ein universelles, kontinuierliches Verteilungsmodell mündet. diff --git a/content/Mathematischer_Hintergrund.tex b/content/Mathematischer_Hintergrund.tex index 76d3310..c37a4de 100644 --- a/content/Mathematischer_Hintergrund.tex +++ b/content/Mathematischer_Hintergrund.tex @@ -1,10 +1,10 @@ %"ltex.language": "de-DE" \subsection{Beweis des zentralen Grenzwertsatzes} -Das Galton-Brett (nach Francis Galton) dient der Veranschaulichung der Binomialverteilung und der experimentellen Bestätigung vom Zentralen Grenzwertsatz im Spezialfall der Binomialverteilung. Im Folgenden formalisieren wir den Weg einer Kugel durch das Brett als stochastischen Prozess mithilfe einer Binomialverteilung und Beweisen anschließend den zentralen Grenzwertsatz von Moivre-Laplace \cite{nahrstedt_monte-carlo-methode_2015}. -\begin{definition}[Modell des Galton Brett]{galton}{nach \cite[S. 252-253]{buchter_elementare_2005}} +Das Galtonbrett (nach Francis Galton) dient der Veranschaulichung der Binomialverteilung und der experimentellen Bestätigung vom Zentralen Grenzwertsatz im Spezialfall der Binomialverteilung. Im Folgenden formalisieren wir den Weg einer Kugel durch das Brett als stochastischen Prozess mithilfe einer Binomialverteilung und Beweisen anschließend den zentralen Grenzwertsatz von Moivre-Laplace \cite{nahrstedt_monte-carlo-methode_2015}. +\begin{definition}[Modell des Galtonbretts]{galton}{nach \cite[S. 252-253]{buchter_elementare_2005}} Sei $(\Omega, \mathcal{F}, P)$ ein Wahrscheinlichkeitsraum. - Der Fall einer Kugel durch ein Galton-Brett mit $n \in \mathbb{N}$ Reihen wird modelliert durch eine Folge von stochastisch unabhängigen und identisch verteilten Zufallsvariablen $X_1, X_2, \dots, X_n$, wobei $X_i \in \{0, 1\}$. + Der Fall einer Kugel durch ein Galtonbrett mit $n \in \mathbb{N}$ Reihen wird modelliert durch eine Folge von stochastisch unabhängigen und identisch verteilten Zufallsvariablen $X_1, X_2, \dots, X_n$, wobei $X_i \in \{0, 1\}$. Dabei beschreibt \textbf{$X_i = 1$ den Fall nach rechts} in der $i$-ten Reihe und \textbf{$X_i = 0$ den Fall nach links}. Die Wahrscheinlichkeit sei $P(X_i = 1) = p$ und $P(X_i = 0) = 1-p = q$. Bei einem symmetrischen Brett gilt $p = q = 0.5$. \end{definition} %Bild? \begin{figure}[htbp] @@ -26,7 +26,7 @@ Das Galton-Brett (nach Francis Galton) dient der Veranschaulichung der Binomialv \end{minipage} \end{figure} \begin{bemerkung}[$X_i$ Bernoulliverteilt]{XiB} - Hierbei ist anzumerken, dass jedes $X_i$ Bernoulliverteilt ist. + Hierbei ist anzumerken, dass jedes $X_i$ bernoulliverteilt ist. \end{bemerkung} Dies gilt, da \[ @@ -38,7 +38,7 @@ Dies gilt, da \end{cases} \] Somit lässt sich auch $X_i \sim \mathcal{B}_{p}$ schreiben.\\ -Um jeden Ausgang des Galton-Brettes durchnummeriert von links nach rechts unterscheiden zu können, definieren wir uns eine weitere Zufallsvariable $S_n$ wie folgt: +Um jeden Ausgang des Galtonbrettes durchnummeriert von links nach rechts unterscheiden zu können, definieren wir uns eine weitere Zufallsvariable $S_n$ wie folgt: \begin{definition}[Zufallsvariable $S_n$]{Sn}{nach \cite[min. 0:50]{statistik_verstehen_beweis_2019}} Die Endposition der Kugel im Fach $k \in \{0, 1, \dots, n\}$ wird durch die Summe der Rechtsabbiegungen beschrieben. Wir definieren die Zufallsvariable: \[S_n = \sum_{i=1}^n X_i\] \end{definition} @@ -50,7 +50,7 @@ Um diese summierte Zufallsvariable genauer zu verstehen betrachten wir zuerst de \] wobei die Zahl $\binom{n}{k}$ die Anzahl der Möglichkeiten, aus einer $n$-elementigen Menge genau $k$-Elemente auszuwählen, angibt. \end{definition} -Betrachten wir nun den Binomialkoeffizienten in Bezug auf $S_n$, beschreibt dieser exakt die Anzahl der möglichen Pfade durch das Galton-Brett, bei denen die Kugel von $n$ Reihen genau $k$-mal nach rechts (und somit $(n-k)$-mal nach links) fällt. Da jeder dieser einzelnen Pfade aufgrund der Unabhängigkeit der Entscheidungen die Wahrscheinlichkeit $p^k (1-p)^{n-k}$ besitzt, ergibt sich die Gesamtwahrscheinlichkeit für das Fach $k$ durch Multiplikation. Daher rührt die Binomialverteilung: +Betrachten wir nun den Binomialkoeffizienten in Bezug auf $S_n$, beschreibt dieser exakt die Anzahl der möglichen Pfade durch das Galtonbrett, bei denen die Kugel von $n$ Reihen genau $k$-mal nach rechts (und somit $(n-k)$-mal nach links) fällt. Da jeder dieser einzelnen Pfade aufgrund der Unabhängigkeit der Entscheidungen die Wahrscheinlichkeit $p^k (1-p)^{n-k}$ besitzt, ergibt sich die Gesamtwahrscheinlichkeit für das Fach $k$ durch Multiplikation. Daher rührt die Binomialverteilung: \begin{definition}[Binomialmodell]{binModell}{\cite{kosenkova_stochastik_2025}} Wiederholt man ein Bernoulli-Experiment mit Erfolgswahrscheinlichkeit $p \in [0,1]$ $n$-mal, und interessiert sich nur für die Anzahl der erfolgreichen Experimente, so wählt man \[ @@ -73,7 +73,7 @@ Anschließend ist folgendes zu bemerken: \end{satz} \begin{proof} Nach \cref{bem:XiB} und \cref{def:Sn} ist $S_n$ die Summe von $n$ unabhängigen Bernoulli-verteilten Zufallsvariablen $X_i \sim \mathcal{B}_p$. Also stellt $S_n$ die Anzahl der Erfolge von n-Wiederholungen von unabhängigen identisch Bernoulli-verteilten Zufallsexperimenten dar. - Daraus folgt, dass das Galton-Brett mit der Zufallsvariable $S_n$ eine Binomialmodell \cref{def:binModell} darstellt und somit die Binomialverteilung nach \cref{def:binZähldichte} eine Zähldichte definiert. Daher gilt: $S_n \sim \mathcal{B}in_{n,p}$. + Daraus folgt, dass das Galtonbrett mit der Zufallsvariable $S_n$ eine Binomialmodell \cref{def:binModell} darstellt und somit die Binomialverteilung nach \cref{def:binZähldichte} eine Zähldichte definiert. Daher gilt: $S_n \sim \mathcal{B}in_{n,p}$. \end{proof} Eine weitere in diesem Kontext interessante Verteilung ist die Normalverteilung. \begin{definition}[Normalverteilung] @@ -187,7 +187,7 @@ Für große $n$ wird die direkte Berechnung der Binomialverteilung aufgrund der \ln\left(1 + z\sqrt{\frac{q}{np}}\right) &\sim z\sqrt{\frac{q}{np}} - \frac{\left(z\sqrt{\frac{q}{np}}\right)^2}{2} + O((z\beta)^3)\\ &= z\sqrt{\frac{q}{np}} - \frac{z^2 q}{2np} + O((z\beta)^3) \end{align*} - und für den anderen ln analog: + und für den anderen $\ln$ analog: \begin{align*} \ln\left(1 - z\sqrt{\frac{p}{nq}}\right) &\sim -z\sqrt{\frac{p}{nq}} - \frac{\left(z\sqrt{\frac{p}{nq}}\right)^2}{2} + O((-z\gamma)^3) \\ &= -z\sqrt{\frac{p}{nq}} - \frac{z^2 p}{2nq} + O((z\gamma)^3) &&\text{"-" fällt wegen O weg} @@ -480,13 +480,13 @@ Die Gamma-Funktion berechnet wie folgt die Fakultät in $\mathbb{N}$: \end{satz} \subsection{Galtonbrett} -Das Galton Brett ist ein bekanntes Modell zur Veranschaulichung der Normalverteilung der Stochastik. Es demonstriert visuell, wie aus einer Vielzahl von zufälligen unabhängigen Einzelereignissen eine feste mathematische Gesetzmäßigkeit entsteht. Naiv betrachtet ist es nicht auf dem ersten Blick deutlich, dass aus der eigentlichen Binomialverteilung die gaußsche Glockenkurve der Normalverteilung entsteht. +Das Galtonbrett ist ein bekanntes Modell zur Veranschaulichung der Normalverteilung der Stochastik. Es demonstriert visuell, wie aus einer Vielzahl von zufälligen unabhängigen Einzelereignissen eine feste mathematische Gesetzmäßigkeit entsteht. Naiv betrachtet ist es nicht auf dem ersten Blick deutlich, dass aus der eigentlichen Binomialverteilung die gaußsche Glockenkurve der Normalverteilung entsteht. -Das Galton Brett wurde von dem britischen Naturforscher und Universalgelehrten Sir Francis Galton (1822–1911) erfunden. Galton war ein Cousin von Charles Darwin und interessierte sich stark für die Vererbung menschlicher Eigenschaften, wie Körpergröße oder Intelligenz \cite{buchter_elementare_2005}. Er stellte auch fest, dass diese Eigenschaften in einer Bevölkerung fast immer einer Glockenkurve der Normalverteilung folgten. Galton suchte nach einem mechanischen Weg, um zu demonstrieren, wie die Vermischung vieler zufälliger genetischer Beiträge der Eltern in der nächsten Generation wieder genau dieselbe Verteilung von Mustern hervorbringt \cite{pearl-mackenzie-2018}. Das Brett diente ihm also ursprünglich als biologisches Analogiemodell für die Vererbung, bevor es zu einem Standardwerkzeug der mathematischen Didaktik wurde, um den zentralen Grenzwertsatz zu visualisieren. Das erste Mal hat Sir Francis Galtin das Brett am 17.02.1874 bei einer Vorlesung in der Royal Society vorgestellt \cite{kit-2022}. +Das Galtonbrett wurde von dem britischen Naturforscher und Universalgelehrten Sir Francis Galton (1822–1911) erfunden. Galton war ein Cousin von Charles Darwin und interessierte sich stark für die Vererbung menschlicher Eigenschaften, wie Körpergröße oder Intelligenz \cite{buchter_elementare_2005}. Er stellte auch fest, dass diese Eigenschaften in einer Bevölkerung fast immer einer Glockenkurve der Normalverteilung folgten. Galton suchte nach einem mechanischen Weg, um zu demonstrieren, wie die Vermischung vieler zufälliger genetischer Beiträge der Eltern in der nächsten Generation wieder genau dieselbe Verteilung von Mustern hervorbringt \cite{pearl-mackenzie-2018}. Das Brett diente ihm also ursprünglich als biologisches Analogiemodell für die Vererbung, bevor es zu einem Standardwerkzeug der mathematischen Didaktik wurde, um den zentralen Grenzwertsatz zu visualisieren. Das erste Mal hat Sir Francis Galtin das Brett am 17.02.1874 bei einer Vorlesung in der Royal Society vorgestellt \cite{kit-2022}. -Das grundlegende Prinzip des Galton-Bretts ist über die 150 Jahre hinweg völlig gleichgeblieben, doch seine Anwendung und Form haben sich stark gewandelt. +Das grundlegende Prinzip des Galtonbretts ist über die 150 Jahre hinweg völlig gleichgeblieben, doch seine Anwendung und Form haben sich stark gewandelt. \begin{itemize} \item Der Anfang in den 1870er Jahren mit Galtons originalem Apparate war nicht viel mehr als ein klobiger, hölzerner Kästen mit Glasscheiben. Als "Kugeln" verwendete er oft Bohnen (weshalb es im Englischen auch Bean Machine genannt wurde). \item Im 20ten Jahrhundert fand das mathematische Prinzip schnell Einzug in die Unterhaltungsindustrie. In Fernsehshows wurde es weltberühmt – so etwa als das Spiel "Plinko" in der US-amerikanischen Show The Price Is Right \cite{pearl-mackenzie-2018} oder in der deutschen Adaption "Der Preis ist heiß". - \item Heute gibt es das Galton-Brett als präzise gefertigte Schreibtisch-Gadgets im Taschenformat, bei denen Tausende winziger Stahlkugeln in lasergeschnittenen Kunststoffrillen laufen. Zudem wird das Brett heute in Schulen und Universitäten meistens digital als Computersimulation wie bei \hyperref{https://www.geogebra.org/m/eayfksqh}{}{}{Galton-Brett mit variabler Wahrscheinlichkeit – GeoGebra} oder \hyperref{https://www.arndt-bruenner.de/mathe/scripts/galtonbrett.htm}{}{}{Galtonbrettsimulation} genutzt, bei der man Parameter, wie die Fallwahrscheinlichkeit nach links oder rechts, flexibel verändern kann. -\end{itemize} \ No newline at end of file + \item Heute gibt es das Galtonbrett als präzise gefertigte Schreibtisch-Gadgets im Taschenformat, bei denen Tausende winziger Stahlkugeln in lasergeschnittenen Kunststoffrillen laufen. Zudem wird das Brett heute in Schulen und Universitäten meistens digital als Computersimulation wie bei \hyperref{https://www.geogebra.org/m/eayfksqh}{}{}{Galtonbrett mit variabler Wahrscheinlichkeit – GeoGebra} oder \hyperref{https://www.arndt-bruenner.de/mathe/scripts/galtonbrett.htm}{}{}{Galtonbrettsimulation} genutzt, bei der man Parameter, wie die Fallwahrscheinlichkeit nach links oder rechts, flexibel verändern kann. +\end{itemize} diff --git a/galton.pdf b/galton.pdf index 5370ca056f1c0010a6f4a6611f7bcea2ce7189c3..8e0f8c12f4cacf690ba2ee53d1c28d2140ea8311 100644 GIT binary patch delta 352185 zcmY(qbzGFs`#o%|C2mlPXG` zn#;Zwh>GSm71AOm>!4I-V;|`8qE?PoF@A(EoWa(W8t=CIr0)BPdTK^c!K#T`MFo3g zTO!#lVJObkYN>{|g6w*5IayUzQH}S=#Qs*7G=35Ka~(Ok=|zp(^*Y)xw)|r0hbZB6 zH`+9jHKn+O2_0Ufz@a1S%RBtK8M~Uz(62sj=QZ*lgFWYid{B1}39RBCpSR=}+lZ&3 zDzyGM*;|^=vMj3|tYu8A8nO-FPhW|(?rWD49)Vz8tnf|iu2USHV@&}j=8uqht7acP zW6dqxo9LV`1MX$T+P13Y6jnboK6R?o@F*E5SUVJiiI`9_i$0{GmE+9*^0dY#`kC;> 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