From b643d9cf8224b02ac954ac3031cad80b1b78c006 Mon Sep 17 00:00:00 2001 From: Elias Fierke Date: Tue, 2 Jun 2026 10:50:04 +0200 Subject: [PATCH 01/10] =?UTF-8?q?head:=20m=C3=B6gliche=20=C3=9Cberschrift?= =?UTF-8?q?=20f=C3=BCr=20Galtonbrett-ZGS-Beweis?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- content/Mathematischer_Hintergrund.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/content/Mathematischer_Hintergrund.tex b/content/Mathematischer_Hintergrund.tex index 026cf6b..962be21 100644 --- a/content/Mathematischer_Hintergrund.tex +++ b/content/Mathematischer_Hintergrund.tex @@ -1,7 +1,7 @@ %"ltex.language": "de-DE" Dies ist der zentrale Teil des Dokuments und soll (inhaltlich) den größten Teil des Dokuments ausmachen. -\subsection{Mathematischer Teil auf Hochsuchschulniveau} +\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 als Binomialverteilung und Beweisen anschließend den Zentralen Grenzwertsatz von Moivre-Laplace. \begin{definition}[Modell des Galton Brett]{galton} Sei $(\Omega, \mathcal{F}, P)$ ein Wahrscheinlichkeitsraum. \\ From ee77717c4a674efbee73d21e1cfc23a97610cd23 Mon Sep 17 00:00:00 2001 From: Elias Fierke Date: Tue, 2 Jun 2026 13:18:35 +0200 Subject: [PATCH 02/10] cnt: Stirling-Beweis --- content/Mathematischer_Hintergrund.tex | 54 +++++++++++++++++++++++++- 1 file changed, 53 insertions(+), 1 deletion(-) diff --git a/content/Mathematischer_Hintergrund.tex b/content/Mathematischer_Hintergrund.tex index 43136b8..d91f07e 100644 --- a/content/Mathematischer_Hintergrund.tex +++ b/content/Mathematischer_Hintergrund.tex @@ -96,7 +96,59 @@ Die Gamma-Funktion berechnet wie folgt die Fakultät in $\mathbb{N}$: \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: + \[ + f(u)\approx f(1)+\frac{f''(1)}{2}(u-1)^2 + = -1 - \frac{(u-1)^2}{2}. + \] + + Einsetzen liefert die lokale Approximation + \[ + \Gamma(x)\approx x^x e^{-x}\cdot\int_0^\infty \exp\!\left(-\frac{x}{2}(u-1)^2\right)\,du. + \] + + 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} From 0f484d1d4e450975d06e17e6dc5dbfa38200b0c8 Mon Sep 17 00:00:00 2001 From: Elias Fierke Date: Tue, 2 Jun 2026 13:18:52 +0200 Subject: [PATCH 03/10] pdf: 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u?R-3Q>)O9xso1g;1KzJ>zQFq(xz>Q&(P4;C*m!x_IJgnNew9>~Li`_&yD)?R From 5bf9bc4fe7527f1d5640881efa0c78dfa7296a39 Mon Sep 17 00:00:00 2001 From: Elias Fierke Date: Wed, 3 Jun 2026 08:35:04 +0200 Subject: [PATCH 04/10] =?UTF-8?q?cnt:=20=C3=BCberschrift=20und=20jahr?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- galton.tex | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/galton.tex b/galton.tex index c34faed..4c41e46 100644 --- a/galton.tex +++ b/galton.tex @@ -10,12 +10,12 @@ -\title{Beispiel Titel} +\title{Das Galtonbrett und der zentrale Grenzwertsatz} \author{Fierke, E. \& Janik, T. \& Weidlich, L. \\ Seminar Erweitertes Fachwissen - Mathe Club} -\date{Sommersemester \the\year} +\date{Sommersemester 2026} \begin{document} \begin{textblock*}{3cm}(1cm,1.5cm) @@ -64,4 +64,4 @@ -\end{document} \ No newline at end of file +\end{document} From 2551772639fbf11d9c8cccf436197e415f34970e Mon Sep 17 00:00:00 2001 From: Elias 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=?UTF-8?q?chore:=20pgfplots-referenz=20hinzugef?= =?UTF-8?q?=C3=BCgt?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- galton.tex | 1 + 1 file changed, 1 insertion(+) diff --git a/galton.tex b/galton.tex index 4c41e46..d9527d8 100644 --- a/galton.tex +++ b/galton.tex @@ -2,6 +2,7 @@ \usepackage[a4paper]{geometry} \geometry{top=3cm,bottom=3cm,left=3cm,right=3.5cm} \usepackage[ngerman]{babel} +\usepackage{pgfplots} \usepackage{Matheclub} From 2c05b9ff88c76a9787c6261fcca95f348862ff0e Mon Sep 17 00:00:00 2001 From: Elias Fierke Date: Wed, 3 Jun 2026 15:08:00 +0200 Subject: [PATCH 07/10] =?UTF-8?q?cnt:=20Taylor-Approximation=20hinzugef?= =?UTF-8?q?=C3=BCgt?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- content/Mathematischer_Hintergrund.tex | 129 +++++++++++++++++++++++++ 1 file changed, 129 insertions(+) diff --git a/content/Mathematischer_Hintergrund.tex b/content/Mathematischer_Hintergrund.tex index 8309732..921acc5 100644 --- a/content/Mathematischer_Hintergrund.tex +++ b/content/Mathematischer_Hintergrund.tex @@ -66,6 +66,135 @@ Anhand des Galton-Brettes lässt sich nun wie bereits beschrieben leicht erkenne % Teilsortiert: +\subsection{Taylor-Approximation} +Um eine Taylorpproximation für eine Funktion zu finden, müssen wir zunächst nachweisen, dass die Funktion die wir approximieren wollen, $(n+1)$-mal differenzierbar ist. Da wir unsere Approximation nach dem quadratischen Term abbrechen werden genügt es zu zeigen, dass $\ln(1-x)$ dreimal differenzierbar ist. +\begin{lemma}{dreimalableiten} + $f(x)=\ln(1-x)$ ist mindestens dreimal differenzierbar. +\end{lemma} +\begin{proof} + Zum Beweis berechnen wir die drei Ableitungen. + \begin{align*} + f'(x)&=(\ln{(1-x)})'\\ + &=\frac{1}{1-x}\cdot(1-x)'&&\text{Kettenregel}\\ + &=-\frac{1}{1-x}\\\\ + f''(x)&=(f'(x))'\\ + &=\left(-\frac{1}{1-x}\right)'\\ + &=\frac{(-1)'\cdot(1-x)-(-1)\cdot(1-x)'}{(1-x)^2}&&\text{Quotientenregel}\\ + &=\frac{(1-x)'}{(1-x)^2}&&\text{da } (-1)'\cdot(1-x)=0\\ + &=\frac{1}{(1-x)^2}\\\\ + f'''(x)&=(f''(x))'\\ + &=(\frac{1}{(1-x)^2})'\\ + &=\frac{(-1)'\cdot(1-x)^2-(-1)\cdot\left((1-x)^2\right)'}{\left((1-x)^2\right)^2}&&\text{Quotientenregel}\\ + &=\frac{\left((1-x)^2\right)'}{(1-x)^4}&&\text{da } (-1)'\cdot(1-x)^2=0\\ + &=\frac{-2\cdot(1-x)}{(1-x)^4}&&\text{Kettenregel}\\ + &=\frac{-2}{(1-x)^3} + \end{align*} +\end{proof} +Nun haben wir nachgewiesen, dass die Funktion $\ln(1-x)$ dreimal differenzierbar ist. +\begin{bemerkung} + Mit einem Blick auf die Funktion und wie ihre Ableitungen gebildet werden, ist leicht zu erkennen, dass $\ln(x-1)$ sogar unendlich oft differenzierbar ist. +\end{bemerkung} +Bevor wir die Approximation nachweisen ist eine formale Definition des Satzes von Taylor und dem Taylorpolynom nötig, wobei letzteres aus dem Satz von Taylor folgt. +\begin{satz}[Satz von Taylor\footnote{Enders, J., Analysis II (2024)}]{taylor} + Seien $n\in\mathbb{N}_0, f:[a,b]\rightarrow\mathbb{R}$ eine $(n+1)$-mal differenzierbare Funktion und $x_0\in[a,b]$. + + Dann gilt für alle $x\in[a,b]$ die Darstellung + \[ + f(x)=\sum_{k=0}^{n}\frac{1}{k!}f^{(k)}(x_0)(x-x_0)^k+R_n(x) + \] + Es existiert ein (von $x$ abhängiges $\xi\in I(x_0,x)$), sodass für das Lagrange-Restglied gilt: + \[ + R_n(x)=\frac{1}{(n+1)!}f^{(n+1)}(\xi)(x-x_0)^{n+1} + \] +\end{satz} + +\begin{bemerkung} + Das Lagrange-Restglied ist in dein meisten Fällen eine Funktion vom Grad $n+1$, aufgrund des letzten Faktors. Es kann aber auch sein, dass das Lagrange-Restglied das Nullpolynom ist, je nach Wahl von $\xi$ und der Auswertung von der $n$-ten Ableitung von $f$ an der Stelle $\xi$. + + Dadurch können wir das Restglied auch auffassen als ein Element von $O(x^n)$. Nach der Definition der Landaunotation gilt: für alle $a,b in \mathbb{R}: a*x^n +b$ liegt in $O(x^n)$. Man kann somit das Landausymbol $O$ als Ansammlung von Funktionen verstehen. Interessanterweise gilt auch eine Relation zwischen den einzelnen "Landaumengen". Für alle $n$ in $\mathbb{N}: O(x^n)$ ist in $O(x^{n+1})$. In der Informatik wird diese Notation genutzt, um asymptotische Verhalten von beispielsweise Laufzeiten zu beschreiben. "Algortihmus A braucht asymptotisch so viel Zeit in Abhängigkeit von der Eingabe, wie eine $x^n$-Funktion". + + Die Definition des Taylorpolynoms entspringt direkt dem Satz von Taylor, nur ohne das Lagrange-Restglied. +\end{bemerkung} +\begin{definition}[Taylorpolynom]{taylorpolynom} + \[ + T_{f,x_0,n}(x)=\sum_{k=0}^{n}\frac{1}{k!}\cdot f^{(k)}(x_0)(x-x_0)^k + \] +\end{definition} +\begin{bemerkung} + Der Satz von Taylor liefert eine lokale Approximation an eine Funktion $f$ durch das Taylorpolynom. +\end{bemerkung} +Nun können wir unsere Approximation von $\ln(1-x)=-x-\frac{x^2}{2}+O(x^3)$ zeigen. +\begin{lemma} + \[ + \ln(1-x)=-x-\frac{x^2}{2}+O(x^3) + \] +\end{lemma} +\begin{proof} + Zunächst ist der Definitionsbereich von $\ln(1-x)$ gleicht dem Intervall $(-\infty,1)$. + + Zudem ist die Funktion $\ln(1-x)$ mindestens dreimal differenzierbar. Betrachten wir $x\in(-\infty,1)$ und $x_0=0$. + + Das zweite Taylor-Polynom lautet + \begin{align*} + T_{\ln(1-x),0,2}(x)&=\sum_{k=0}^{2}\frac{1}{k!}\cdot (\ln(1-x_0))^{(k)}(x-x_0)^k\\ + &=\frac{1}{0!}\cdot\ln(1)\cdot 1+\frac{1}{1!}\cdot(\ln(1))^{(1)}\cdot x+\frac{1}{2!}\cdot(\ln(1))^{(2)}\cdot x^2&&\text{\cref{lem:dreimalableiten}}\\ + &=0+1\cdot(-1)\cdot x+\frac{1}{2}\cdot(-1)\cdot x^2\\ + &=-x-\frac{x^2}{2} + \end{align*} + Aus dem Satz von Taylor (\cref{satz:taylor}) folgt somit: + \begin{align*} + \ln(1-x)&=-x-\frac{x^2}{2}+R_2(x) + \end{align*} + wobei das Lagrange-Restglied etwa $R_2(x)\approx O(x^3)$, denn + \begin{align*} + R_2(x)&=\frac{1}{(2+1)!}\cdot(\ln(1-\xi))^{(2+1)}\cdot(x-0)^{(2+1)}\\ + &=-\frac{1}{3(1-\xi)^3}\cdot x^3\quad\text{für ein }\xi\in(0,x)&&\text{\cref{lem:dreimalableiten}}\\ + \end{align*} + wobei nach der Landau-Notation gilt: $a\cdot x^n+b\in O(x^n)$ + \[ + \Rightarrow\quad \frac{1}{3(1-\xi)^3}\cdot x^3\in O(x^3), + \] + da $\frac{1}{3(1-\xi)^3}$ konstant für ein $\xi$ im Intervall. + + Da $\xi$ variabel ist (also von $x$ abhängt), steht $O(x^3)$ stellvertretend für ein gut passendes Polynom vom Grad 3. + \[ + \Rightarrow\quad\ln(1-x)\approx -x -\frac{x^2}{2}+O(x^3) + \] +\end{proof} +\begin{bemerkung} + Aus der Definition folgt die Gleichheit beider Funktionen, aber genauer betrachtet kann die ln-Funktion nicht durch ein Polynom dargestellt werden. Es wird immer einen gewissen Fehler geben. Da auch in abhängigkeit von $x$ das $\xi$ gewählt wird, handelt es sich bei dem Lagrange-Restglied nicht um ein Polynom, weil es sich dynamisch verändert. + + Wenn wir uns beide Funktionen einmal in GeoGebra plotten, können wir sehen, dass beide Funktionen um unsere Entwicklungsstelle $x_0=0$ herum sehr ähnlich sind. +\end{bemerkung} +\begin{figure}[H] + \centering + \begin{tikzpicture} + \begin{axis}[ + domain=-2.95:2.95, + samples=400, + axis lines=middle, + axis equal, + xmin=-3, + xmax=3, + ymin=-3.5, + ymax=3.5, + xtick distance=1, + ytick distance=1, + legend pos=south west, + grid=both + ] + % ln(1-x) + \addplot[blue, thick] {ln(1-x)}; + \addlegendentry{$\ln(1-x)$} + + % Näherung -x - x^2/2 + \addplot[red, thick, dashed] {-x - x^2/2}; + \addlegendentry{$-x-\frac{x^2}{2}$} + \end{axis} + \end{tikzpicture} + \caption{Darstellung der Funktionen $\ln(1-x)$ und $-x-\frac{x^2}{2}$} +\end{figure} +Würden wir die Entwicklung weiterführen, approximiert die violette Funktion (unser Taylorpolynom) die orangene Funktion $(\ln(1-x))$ immer weiter. Graphisch lässt sich erkennen, dass für Werte im Intervall $I(-0.5,0.5)$ die Funktion $\ln(1-x) \approx -x\frac{-x}{2}$ plus ein kleiner Fehler ist. \subsection{Gamma-Funktion} % wird für Beweis der Stirling-Approximation gebraucht Zum nachfolgenden Betrachtung und des Beweises der Stirling-Approximation betrachten wir nun die Gamma-Funktion, auch \textbf{Eulersches Integral zweiter Gattung} genannt. Sie erweitert die Fakultätsfunktion von den natürlichen Zahlen $\mathbb{N}$ auf reelle und komplexe Zahlen (mit einigen Ausnahmen). Hier betrachten wir der Einfachheit halber nur die Gamma-Funktion in $\mathbb{R}$. 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15:17:51 +0200 Subject: [PATCH 09/10] fix: Formulierung Plot --- content/Mathematischer_Hintergrund.tex | 3 +-- 1 file changed, 1 insertion(+), 2 deletions(-) diff --git a/content/Mathematischer_Hintergrund.tex b/content/Mathematischer_Hintergrund.tex index 921acc5..27b26ef 100644 --- a/content/Mathematischer_Hintergrund.tex +++ b/content/Mathematischer_Hintergrund.tex @@ -163,9 +163,8 @@ Nun können wir unsere Approximation von $\ln(1-x)=-x-\frac{x^2}{2}+O(x^3)$ zeig \end{proof} \begin{bemerkung} Aus der Definition folgt die Gleichheit beider Funktionen, aber genauer betrachtet kann die ln-Funktion nicht durch ein Polynom dargestellt werden. Es wird immer einen gewissen Fehler geben. Da auch in abhängigkeit von $x$ das $\xi$ gewählt wird, handelt es sich bei dem Lagrange-Restglied nicht um ein Polynom, weil es sich dynamisch verändert. - - Wenn wir uns beide Funktionen einmal in GeoGebra plotten, können wir sehen, dass beide Funktionen um unsere Entwicklungsstelle $x_0=0$ herum sehr ähnlich sind. \end{bemerkung} +Im Plot können wir sehen, dass beide Funktionen um unsere Entwicklungsstelle $x_0=0$ herum sehr ähnlich sind. \begin{figure}[H] \centering \begin{tikzpicture} From 1dc7c0ca590be7adc61e66f422888414c822e097 Mon Sep 17 00:00:00 2001 From: Elias Fierke Date: Wed, 3 Jun 2026 15:18:15 +0200 Subject: [PATCH 10/10] pdf: neue Version mit fix --- galton.pdf | Bin 235674 -> 235633 bytes 1 file changed, 0 insertions(+), 0 deletions(-) diff --git a/galton.pdf b/galton.pdf index 119a687ea9abddcfef54c387a8513b19fa80fdef..d8a0304401580dd77b805ad92f8d7a6685dac6a2 100644 GIT binary patch delta 3768 zcmV;p4oC5tu@3RD4zQ$$0yQz0u~Y*nf5lo`kJ~mDexF~Vee<9Z!@H<03hWkbf)+)$ zN%OEk+6QH<*hXwk>r0aMW&iqqhoo#tL&=`Wqy;9C%*!Ep`0gwq;Ei%Db$xD&FdYd`JxGJ+k8|J8Fp1q#^!VSOv_0{)JpV^|@mTEW`vC8(Vf4&UG zed(`rVT+l7z}QR6ezj6+c4+2Fk-qX{>}Rurl?C@T)-h9-SuU7WHcXkPMlV=lRn}?s z)paflcWm6B%ZyQRmKT;ODjqef%mhtOA}T1gX=m z$9Y)Sm6IEQLqPRR7IBmRn^+rX!W2-wVugs}-lZ0c`OLmbf>3g%g`!8Hf8s!1;aA_j zy88V};9T+y9ZJnOHrbYr$QZU^ffnSa%e<(vr>36Y*%7NvGjd*&ieQgS#+WE!$ zlU_`mbPeUotka4>(t&nYf2PBl40;%@hX@?NdUU5dlEv^mbDQN!;Un;V97QS4xHgw5 zN(~j2SiEKu=CQMmBz|&XG%F3pRRvR8w^o+ysDKD*AX2$;)WganBSC1xhIu)b(gzYZ03a7{nCoy#(67vfRmxYJ@q{0S&Y&8G4j5_V#{h+FdV`Bh|ruq8~y zCV|S{F-6h}_-e31e+edAxB=mdVDOIzAL40VoMtN*Ta5<+8=k)4XMimY@QSWhmCO{y z6tD^T)0U)_B}uP)GA|jtnicX~#0H#5aQPC#sf!TM# z`PQ2d1?~sX@)J4zUr4HRMqNAQhi!COt7gnof9Z~{hI&?jryO$z&f}4f-Bb8GiF-Qf z*ST?}I3KF&M5r3zuR(a>c7$`|4!)^wmchf%>!_a}s%Bmt@*i*a)qd6T;Kgd)Qqz=! zug0*cE<{AxBS=^gpx<$VO-grwsO`DR2_p7p{OHTiA$c>{blAcF+UKTowiI!@j1NRom=vvoUJxN#COfU-oCHa>v&yV zQpir1;m(KA@w$3s6gM@BR($r}XIHkXDmtb$26bw-+R@5g$h1`J-EH_qiJ8%Xe?E?^ z1y$9XWr*I50!U3lg`5evlTIac>|l1{FFIsQk;~un*hyTeaIkZoLaVm9WhCm{KvL;|%Y7rttnYBJBCL ziunY^(pMJB?@lg2j?a0!*SR43fB6{`WYejH$Ip=oxEHC#1xYe}E=dLjDhE@@jSu5L zElDQ9I{`LWaggxhPLk}x9NA+zI=Yai!0Ew`uCFTXzzYCfiSuF5b)YI$x}&N%?F>Ta zpsBtHqa8%Rhuuv{c17U0=#vXeiGT|$A0&bDCO9+qUPmh+ey=4U$AZ3~fB)|pB8}&n z4>zv72m={ZjK8)U_m|Gf4=5PQ;=k_u%_a-N`X7zG=t{{?VqU}nFzVCa7H|J`Q<1R6 zjwZq3WSv3?j&=B-^vRaC(qT3JcL4)0g!i-3LJtU#l;fp7_B$5bWLKIaZYs6xl&-0? 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