From d35e2db4028484fad0a05114ed6b3f748f607599 Mon Sep 17 00:00:00 2001 From: Luca Weidlich Date: Wed, 1 Jul 2026 14:30:08 +0200 Subject: [PATCH] =?UTF-8?q?einige=20=C3=A4nderungen=20und=20erg=C3=A4nzung?= =?UTF-8?q?en=20allgemein=20und=20pascallsches=20Dreieck=20raus?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- content/Kernideen.tex | 12 +- content/Mathematischer_Hintergrund.tex | 166 +++++++++---------------- content/Stichworte.tex | 11 +- content/Weiterfuehrende_Ideen.tex | 5 +- galton.pdf | Bin 461075 -> 458000 bytes 5 files changed, 73 insertions(+), 121 deletions(-) diff --git a/content/Kernideen.tex b/content/Kernideen.tex index 3ebc8bf..219bc37 100644 --- a/content/Kernideen.tex +++ b/content/Kernideen.tex @@ -1,7 +1,11 @@ -Ziel des Kapitels ist es, der Leserschaft auf einen Blick die zentralen Ideen zu präsentieren. +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. -Beschreiben Sie hier also knapp die mathematische Fragestellung. Es soll klar der mathematische Rahmen umrissen werden und die Fragestellung in den größeren Kontext eingeordnet werden. +%Informell lautet unsere Fragestellung: Lässt sich das exakte, aber für große Zahlen unhandliche Zählverfahren der Kugelwege (Kombinatorik) durch eine einfachere, kontinuierliche Kurve beschreiben? -Falls es einen zentralen Satz gibt, der die Fragestellung beantwortet, dann kann dieser hier auch genannt werden (auch wenn er später nochmal auftaucht und bewiesen wird). +Der mathematische Rahmen bewegt sich dabei im Übergang von der diskreten Wahrscheinlichkeitstheorie zur stetigen Analysis. Wir modellieren den Fall einer Kugel zunächst als Summe unabhängiger, identisch verteilter Bernoulli-Zufallsvariablen, was direkt zur Binomialverteilung führt. Da die direkten Berechnungen der Binomialkoeffizienten für eine große Anzahl an Reihen (für $n \to \infty$) aufgrund der Fakultäten extrem aufwendig werden, suchen wir nach einer asymptotischen Näherung. Außerdem ist es ein sehr schönes Resultat, dass aus einer Binomialverteilung eine Normalverteilung "entstehen" kann. -Ansonsten soll die Fragestellung zwar klar umrissen werden, aber eher informell beschrieben werden. \ No newline at end of file +Den größeren Kontext bildet die Verknüpfung grundlegender mathematischer Werkzeuge, um genau dies zu beweisen. Wir greifen auf die \emph{Stirling-Approximation} (zur Näherung von Fakultäten) und die \emph{Taylor-Entwicklung} (zur Approximation von Logarithmen) zurück. Diese analytischen Werkzeuge schlagen die Brücke zwischen der diskreten Welt des Pascalschen Dreiecks und der kontinuierlichen Welt der Exponentialfunktion. + +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 diff --git a/content/Mathematischer_Hintergrund.tex b/content/Mathematischer_Hintergrund.tex index 7c19a5c..3832ecf 100644 --- a/content/Mathematischer_Hintergrund.tex +++ b/content/Mathematischer_Hintergrund.tex @@ -265,57 +265,70 @@ Folgende Proposition ist leicht aus dem oberen Beweis ableitbar (aufgrund der Ab wobei $\beta = \sqrt{\frac{q}{np}}$ und $\gamma = \sqrt{\frac{p}{nq}}$ definiert sind. \end{proposition} - -% Teilsortiert: -\subsection{Pascalsches Dreieck} -Das Pascalsche Dreieck ist ein geometrisches Dreieck aus Zahlen, das sich unendlich nach unten fortsetzt und dabei die Binomialkoeffizienten repräsentiert. Aus \cref{def:binomialkoeffizienten} entwickeln wir das Pascalsche Dreieck. -\begin{definition}[Pascalsches Dreieck]{pascaldreieck}{\cite{berger2023-mz}} - Das \textbf{Pascalsche Dreieck} ist das unendliche Zahlenschema +\subsection{Stirling-Approximation} +Die Stirling-Approximation ist eine mathematische Näherung zur Berechnung der Fakultät einer Zahl. Die Approximation ist vorallem in der Stochastik und der statistischen Physik ein unverzichtbares Werkzeug. Wir schauen sie daher im Folgenden im Detail an. +\begin{satz}[Stirlingformel]{stirlingformel}{\cite{freitag1995-oj}} + Für $n\mapsto\infty$ gilt nach \textsc{Stirlings} Approximation, dass \[ - P=(p_{n,k})_{n\in\mathbb{N}_0,0\leq k\leq n} + n!\sim \sqrt{2\pi n}\left(\frac{n}{e}\right)^n \] - mit + Das bedeutet \[ - p_{n,k}=\binom{n}{k} - \] -\end{definition} -Die ersten Zeilen des Pascalschen Dreiecks lauten: -\[ -\begin{array}{ccccccccc} - &&&&1\\ - &&&1&&1\\ - &&1&&2&&1\\ - &1&&3&&3&&1\\ - 1&&4&&6&&4&&1 -\end{array} -\] -\begin{satz}[Pascalsche Rekursion] - Für $n\geq1$ und $1\leq k\leq n-1$ gilt: - \[ - \binom{n}{k} = \binom{n-1}{k-1}+\binom{n-1}{k} + \lim_{n\rightarrow\infty}\frac{n!}{\sqrt{2\pi n}\left(\frac{n}{e}\right)^n}=1 \] \end{satz} -\begin{bemerkung} - Diese Beziehung erklärt die Entstehung jeder inneren Zahl als Summe der beiden darüberliegenden Zahlen. -\end{bemerkung} \begin{proof} - Sei $M$ eine Menge mit $n$ Elementen. Wähle $x\in M$ fest. - - Wir zählen die $k$-elementigen Teilmengen von M. - \begin{itemize} - \item Enthält eine Teilmenge das Element $x$, dann müssen noch $k-1$ Elemente aus den $n-1$ verbleibenden Elementen gewählt werden. Wir erhalten - \[ - \binom{n-1}{k-1} - \] - \item Enthält eine Teilmenge das $x$ nicht, werden $k$ Elemente aus den verbleibenden $n-1$ Elementen gewählt, also - \[ - \binom{n-1}{k} - \] - \end{itemize} - Aufrund der Disjunkteit beider Fälle und der Erfassung aller $k$-elementigen Teilmengen, folgt + Der Ausgangspunkt ist die nach \cref{def:gammafunktion} definierte \textbf{Gamma-Funktion} \[ - \binom{n}{k} = \binom{n-1}{k-1}+\binom{n-1}{k} + \Gamma(x)=\int_{0}^{\infty}t^{x-1}e^{-t} dt \] + Wir schreiben den Integranden als Exponentialfunktion, dann haben wir + \[ + t^{x-1}e^{-t} = \exp((x-1)\ln t-t) + \] + also ist + \[ + \Gamma(x)=\int_{0}^{\infty}\exp((x-1)\ln t-t) dt + \] + Setze $t=xu$ und $f(u)=\ln u - u$, dann erhalten wir + \begin{align} + \Gamma(x)&=x^x\cdot\int_{0}^{\infty}u^{x-1}e^{-xu} du\\ + &=x^x\cdot\int_{0}^{\infty}\exp(x(\ln u-u)) du\\ + &=x^x\cdot \int_{0}^{\infty} e^{xf(u)}du + \end{align} + + Für große \(x\) wird das Integral durch die Umgebung des kritischen Punktes von \(f\) dominiert. Diese erhält man aus + \[ + f'(u)=\frac{1}{u}-1=0 \quad \Rightarrow \quad u=1. + \] + + Weiter gilt + \[ + f''(u)=-\frac{1}{u^2}, \quad \text{also } f''(1)=-1. + \] + + Damit besitzt \(f\) bei \(u=1\) einen stationären Punkt, und wir entwickeln \(f\) dort bis zur zweiten Ordnung: + \[ + f(u)\sim f(1)+\frac{f''(1)}{2}(u-1)^2 + = -1 - \frac{(u-1)^2}{2}. + \] + + Einsetzen liefert die lokale Approximation + \[ + \Gamma(x)\sim x^x e^{-x}\cdot\int_0^\infty \exp\!\left(-\frac{x}{2}(u-1)^2\right)\,du. + \] + + Da der Hauptbeitrag aus einer Umgebung von $u=1$ stammt, kann das Integral asymptotisch auf $\mathbb{R}$ erweitert werden. + \[ + \int_{-\infty}^{\infty} \exp\!\left(-\frac{x}{2}(u-1)^2\right)\,du + = \sqrt{\frac{2\pi}{x}}. + \] + + Damit erhält man + \[ + \Gamma(x)\sim \sqrt{2\pi}\, x^{x-\frac12} e^{-x}. + \] + \end{proof} \subsection{Taylor-Approximation} @@ -465,71 +478,4 @@ Die Gamma-Funktion berechnet wie folgt die Fakultät in $\mathbb{N}$: \] \end{satz} -\subsection{Stirling-Approximation} -Die Stirling-Approximation ist eine mathematische Näherung zur Berechnung der Fakultät einer Zahl. Die Approximation ist vorallem in der Stochastik und der statistischen Physik ein unverzichtbares Werkzeug. Wir schauen sie daher im Folgenden im Detail an. -\begin{satz}[Stirlingformel]{stirlingformel}{\cite{freitag1995-oj}} - Für $n\mapsto\infty$ gilt nach \textsc{Stirlings} Approximation, dass - \[ - n!\sim \sqrt{2\pi n}\left(\frac{n}{e}\right)^n - \] - Das bedeutet - \[ - \lim_{n\rightarrow\infty}\frac{n!}{\sqrt{2\pi n}\left(\frac{n}{e}\right)^n}=1 - \] -\end{satz} -\begin{proof} - Der Ausgangspunkt ist die nach \cref{def:gammafunktion} definierte \textbf{Gamma-Funktion} - \[ - \Gamma(x)=\int_{0}^{\infty}t^{x-1}e^{-t} dt - \] - Wir schreiben den Integranden als Exponentialfunktion, dann haben wir - \[ - t^{x-1}e^{-t} = \exp((x-1)\ln t-t) - \] - also ist - \[ - \Gamma(x)=\int_{0}^{\infty}\exp((x-1)\ln t-t) dt - \] - Setze $t=xu$ und $f(u)=\ln u - u$, dann erhalten wir - \begin{align} - \Gamma(x)&=x^x\cdot\int_{0}^{\infty}u^{x-1}e^{-xu} du\\ - &=x^x\cdot\int_{0}^{\infty}\exp(x(\ln u-u)) du\\ - &=x^x\cdot \int_{0}^{\infty} e^{xf(u)}du - \end{align} - - Für große \(x\) wird das Integral durch die Umgebung des kritischen Punktes von \(f\) dominiert. Diese erhält man aus - \[ - f'(u)=\frac{1}{u}-1=0 \quad \Rightarrow \quad u=1. - \] - - Weiter gilt - \[ - f''(u)=-\frac{1}{u^2}, \quad \text{also } f''(1)=-1. - \] - - Damit besitzt \(f\) bei \(u=1\) einen stationären Punkt, und wir entwickeln \(f\) dort bis zur zweiten Ordnung: - \[ - f(u)\sim f(1)+\frac{f''(1)}{2}(u-1)^2 - = -1 - \frac{(u-1)^2}{2}. - \] - - Einsetzen liefert die lokale Approximation - \[ - \Gamma(x)\sim x^x e^{-x}\cdot\int_0^\infty \exp\!\left(-\frac{x}{2}(u-1)^2\right)\,du. - \] - - Da der Hauptbeitrag aus einer Umgebung von $u=1$ stammt, kann das Integral asymptotisch auf $\mathbb{R}$ erweitert werden. - \[ - \int_{-\infty}^{\infty} \exp\!\left(-\frac{x}{2}(u-1)^2\right)\,du - = \sqrt{\frac{2\pi}{x}}. - \] - - Damit erhält man - \[ - \Gamma(x)\sim \sqrt{2\pi}\, x^{x-\frac12} e^{-x}. - \] - -\end{proof} - - diff --git a/content/Stichworte.tex b/content/Stichworte.tex index 4896e77..189b9ad 100644 --- a/content/Stichworte.tex +++ b/content/Stichworte.tex @@ -2,13 +2,12 @@ Die folgenden \textbf{Stichworte} decken in etwa den Inhalt des ausgearbeiteten \begin{itemize} \item Stochastik - \item Welche fundamentalen Ideen werden behandelt? - \item zentraler Grenzwertsatz als "Mittelpunkt" - \item Binomial- zu Normalverteilung - \item Satz von de Moivre und Laplace - \item Pascalsches Dreieck - \item Kreiszahl $\pi$ + \item Binomialverteilung + \item Normalverteilung + \item Zentraler Grenzwertsatz von de Moivre-Laplace \item Stirling-Approximation + \item Taylor-Entwicklung + \item Pascalsches-Dreieck \item Gamma-Funktion \item Klasse 10 bis 12 \end{itemize} \ No newline at end of file diff --git a/content/Weiterfuehrende_Ideen.tex b/content/Weiterfuehrende_Ideen.tex index fbacc58..ba20b6c 100644 --- a/content/Weiterfuehrende_Ideen.tex +++ b/content/Weiterfuehrende_Ideen.tex @@ -1 +1,4 @@ -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! \ No newline at end of file +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 diff --git a/galton.pdf b/galton.pdf index bca6edfc17052dd76b08c95c315cc312d04135a6..1efcf823fc6099397cb2cb2ee1b69e0f06594a9d 100644 GIT binary patch delta 138997 zcmZ^~Wmp_-v#w1BcXti$Ft`)k-Q6X)I}8@wf(=e^cXx;2?!kghA8NyXU;RCMl69;fb*J2_7th34;wf*B!T^;zrweW_W^?q)6Fh#q&FSLK$W5 z>qyWb(?S0D_$HyASgx*m@6WR*LWZ~7tV_Sx5hq*KV#L+u?>*=FnzNZBWGMebUfsCU zb$xwWskn1Ca}|j)C22h$#=>u9v)e(~Z;rsQi7U4WpN0o<0{DS&>2}IS0EDgBNh#wj3r9JeddGXW@yuwyIgIsq?&UNtrlgE)B7X6Bzs4GD@Z<@!M>D zH4tkVY3WYXT!Q$L+pDH~E7}c}b$ol#yuQaumV`L01|t z4?uiL*a_;85NtJ}!G)p1rK2D6UuW)ln`?P~MdWi>iP6*n=Td0;DUVA19Kxl-@o;OQ zU0p9)&Bsc*TWGP@3AXO2)R;LAd;lgfC{5Apr6##6P9$BE7f%gLpI;_rzMR1c52Ko6 ze`)Dt6+YrAMg5Es6Bs&|ap&m;dbQt7RUp})go4fhuL`&|MaLg}=a2+S`ZcJnXD}Fr z-&!E8tAVY}_cK)|&#)IaQpcv;^GG?; 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