From 5c8ff7b3a7d5f9c8829a1119b8374e5ceb36b4c0 Mon Sep 17 00:00:00 2001 From: Luca Weidlich Date: Sun, 12 Jul 2026 16:04:59 +0200 Subject: [PATCH 1/5] Weiteres Dings Kapitel und kleinigkeiten --- bibliography.bib | 28 +++++++++++++++++++++++++ content/Mathematischer_Hintergrund.tex | 2 +- content/Weiterfuehrende_Ideen.tex | 13 ++++++++---- galton.pdf | Bin 493249 -> 479415 bytes 4 files changed, 38 insertions(+), 5 deletions(-) diff --git a/bibliography.bib b/bibliography.bib index 7f6d858..8a1fdce 100644 --- a/bibliography.bib +++ b/bibliography.bib @@ -133,3 +133,31 @@ address = {Humboldt-Universität zu Berlin}, url = {https://www2.informatik.hu-berlin.de/~koessler/Stochastik/Stochastik2006/FolienStochastik_4.pdf} } + + +@article{schreiter_echt_2022, + title = {Echt schräg: Galton revisited}, + url = {https://www.friedrich-verlag.de/friedrich-plus/sekundarstufe/mathematik/stochastik/echt-schrag-galton-revisited-12235}, + number = {232}, + publisher = {Friedrich Verlag {GmbH}}, + author = {Schreiter, Saskia and Vogel, Markus and Eichler, Andreas}, + urldate = {2026-07-12}, + date = {2022}, +} + + +@incollection{posedel_martingale_2025, + location = {Berlin, Heidelberg}, + title = {Martingale Central Limit Theorem}, + isbn = {978-3-662-69359-9}, + url = {https://doi.org/10.1007/978-3-662-69359-9_345}, + doi = {10.1007/978-3-662-69359-9_345}, + abstract = {The martingale central limit theorem ({MCLT}) links the notions of martingales and the Lindeberg-Feller classical central limit theorem ({CLT}) for independent summands.}, + pages = {1418--1421}, + booktitle = {International Encyclopedia of Statistical Science}, + publisher = {Springer Berlin Heidelberg}, + author = {Posedel, Petra}, + editor = {Lovric, Miodrag}, + date = {2025}, +} + diff --git a/content/Mathematischer_Hintergrund.tex b/content/Mathematischer_Hintergrund.tex index f6bcbd1..3309762 100644 --- a/content/Mathematischer_Hintergrund.tex +++ b/content/Mathematischer_Hintergrund.tex @@ -88,7 +88,7 @@ Für große $n$ wird die direkte Berechnung der Binomialverteilung aufgrund der \[ P(S_n = k) \sim \mathcal{N}(\mu,\sigma ^2) \] - wobei $\mathbb{E}(X)=\mu$, $Var(X)=\sigma ^2$ und $\frac{k-np}{\sqrt{npq}}$ begrenzt ist. + wobei $\mathbb{E}(X)=\mu$, $Var(X)=\sigma ^2 < \infty$ und $\frac{k-np}{\sqrt{npq}}$ begrenzt ist. \end{satz} \begin{proof} Der Beweis folgt zum großen Teil dem Beweis von \cite{statistik_verstehen_beweis_2019} und wurde von den Autoren weiter konkretisiert.\\ diff --git a/content/Weiterfuehrende_Ideen.tex b/content/Weiterfuehrende_Ideen.tex index ba20b6c..2802f66 100644 --- a/content/Weiterfuehrende_Ideen.tex +++ b/content/Weiterfuehrende_Ideen.tex @@ -1,4 +1,9 @@ -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 +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 wie praktische Erweiterungsmöglichkeiten, die weit über das Standardmodell hinausgehen. +\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. Unter bestimmten Voraussetzungen greif hier dann der Martingal-Zentralgrenzwertsatz, wodurch wieder die Normalverteilung angenähert wird \cite{posedel_martingale_2025}. + +\paragraph{Identisch Verteilt} +Das klassische Brett nutzt auf jeder Ebene die identische Wahrscheinlichkeit $p = 0.5$. Da auch dies nicht unbedingt immer Gewehrleitbar ist, lässt sich diese Bedingung ebenso aufweichen. Unter der Bedingung, dass keine einzelne Reihe die Gesamtvarianz dominiert (Lindeberg-Bedingung), führt der verallgemeinerte zentrale Grenzwertsatz dennoch zu einer Normalverteilung. Die Glockenkurve bleibt erhalten, auch wenn die Elementarschritte auf dem Weg nach unten variieren \cite{posedel_martingale_2025}. + +\paragraph{Erweiterung der Mechanik in der Schule: Das schiefe Galtonbrett} +Eine physisch leicht umzusetzende Variation ist das Neigen des gesamten Brettes. Durch die Schräglage wird die Wahrscheinlichkeit für einen Fall nach rechts oder links asymmetrisch, sodass $p \neq 0.5$ gilt. Die Pfade der Kugeln folgen nun einer asymmetrischen Binomialverteilung. Für eine große Anzahl an Stiftreihen $n$ nähert sich die Form (nach dem Satz von Moivre-Laplace) zwar immer noch einer Normalverteilung an, jedoch verschieben sich der Erwartungswert auf $\mu = np$ und die Varianz auf $\sigma^2 = np(1-p)$. 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zBWsr@nq>DDN~ejLjjabs3v&$i$ikv%S~=(uaPqbx>^>4JQVL<1?tv`= z%P~}r6MZCft@ys9oKeBi=xdng9EIGdWe{W&{{Y*_{BXzEuHob8N$Gmc?k>^0YRKuf z@jEZb=?VNDi73yhz2{W&%;U0J_+FX5r7rF zLFW*`ExoJV@84-S!Y@Z(Qo%1|pOPIxCw*YtUv{1}y1`ewlnT8NbN-#5l)L9Erh!mh zP4MG`oOl>4Xtm1XbDQ=)U}KF4=0-!Vu@Td##Avxwn_){b0LxH=W*SML*el$Gb`M8; zOz^~}BoXB=W&9c>I$_}_*5HarCyBG?e-h>XJ1!Iy>=YOr7I@=oK}yn!GEx+Rg1Y8< G6#oGv+gjED From 9cb7744894bd57514408c2e77be97551caa8274f Mon Sep 17 00:00:00 2001 From: Luca Weidlich Date: Sun, 12 Jul 2026 17:15:05 +0200 Subject: [PATCH 2/5] =?UTF-8?q?L=C3=B6sung=20AB3?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- content/Arbeitsmaterialien.tex | 7 +- images/AB3_GaltonVersuche_L.png | Bin 0 -> 136431 bytes images/AB3_Galton_L.png | Bin 0 -> 48526 bytes images/AB3_X_L.png | Bin 0 -> 11276 bytes material/ab3.pdf | Bin 275904 -> 274305 bytes material/ab3.tex | 22 +- material/ab3_Lös.pdf | Bin 0 -> 342493 bytes material/ab3_Lös.tex | 442 ++++++++++++++++++++++++++++++++ 8 files changed, 464 insertions(+), 7 deletions(-) create mode 100644 images/AB3_GaltonVersuche_L.png create mode 100644 images/AB3_Galton_L.png create mode 100644 images/AB3_X_L.png create mode 100644 material/ab3_Lös.pdf create mode 100644 material/ab3_Lös.tex diff --git a/content/Arbeitsmaterialien.tex b/content/Arbeitsmaterialien.tex index 00a73db..b1dba69 100644 --- a/content/Arbeitsmaterialien.tex +++ b/content/Arbeitsmaterialien.tex @@ -31,4 +31,9 @@ Bitte bedenken Sie, zumindest Lösungsskizzen zu den Aufgaben zu geben! Besser n % 2 -> Vorfaktor "abschätzen" % 3 -> potenzen Killen um E funktion zu gewinnen! % ln und umformen (Klassenstufe! -> Anschätzung und Ergebnis vorgeben, nicht rechnen) - % Rücksubstitution machen \ No newline at end of file + % Rücksubstitution machen +Das dritte Arbeitsblatt dient dem Beweis des zentralen Grenzwertsatzes von Movire-Laplace. Ziel ist es hierbei explizit nicht, dass die Schüler*innen den Beweis in jedem Detail, wie im Teil "Mathematischer Hintergrund", verstehen aber die Grundlegenden Argumentationen und mathematischen Tricks nachvollziehen. Außerdem ist vor allem das Beweisen etwas, dass im Rahmenlehrplan (\cite{lisum-2022} und \cite{lisum-2023}) nicht gefordert ist, wodurch es sich sehr gut für eine MatheAG oder Zusatzstunde handelt, in der Inhalte über den klassischen Unterricht hinaus behandelt werden.\\ +Für das Arbeitsblatt sollten die Schüler*innen bereits mit der Binomialverteilung, Zufallsvariablen, Logarithmen sowie der e-Funktion vertraut sein und mit etwas Formalismus umgehen können. Tiefgreifende Kenntnisse zur Normalverteilung sind nicht vorausgesetzt und werden auf dem Arbeitsblatt in ausreichender Form wiederholt bzw. können selbständig aufbereitet werden. Es ist empfohlen mindestens eines der Arbeitsblätter Galton x Pascall oder Galton x Plinko vorher bearbeitet zu haben, um das Konzept des Galtonbrettes und dessen Zusammenhang mit der Binomialverteilung verstanden zu haben.\\ +Auf dem Arbeitsblatt wird die formale Modellierung des Galtonbrettes vorgestellt und soll durch eine Zuordnungsaufgabe durch die Schüler*innen vertieft werden. Im Anschluss soll der Bezug zwischen Binomial- und Normalverteilung wiederholt bzw. erarbeitet werden. Das Kernstück des Arbeitsblattes stellt der Lückentext-Ähnliche Beweis dar. Hier sollen die Schüler einige Mathematische Argumentationen nachvollziehen und einige Umformungen selbst durchführen, um den Beweis nachvollziehen zu können. 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zlG~ffY@@I*IwiNFR~J*-((aZ2BXS*oX)t?0Z}9i_`?qFwz}_N{&YnAW@yctn{{hx) B=Jo&p diff --git a/material/ab3.tex b/material/ab3.tex index 187bce6..14adeaf 100644 --- a/material/ab3.tex +++ b/material/ab3.tex @@ -298,24 +298,29 @@ Diesen Satz wollen wir nun Semi-Formal beweisen.\\ \textbf{\underline{Aufgabe 3:}} Fülle die Lücken im Beweis aus. Wenn du dir unsicher bist kannst du dir alle fehlenden Textbausteine auf dem Zusatzblatt ansehen. \begin{enumerate} + \item Bevor wir den Satz beweisen, ist es wichtig, die Aussage zu verstehen. Formuliere die Kernaussage den Satzes kurz in eigenen Worten: + \answer{3} \item Zuerst muss uns nochmal klar werden, was unser Ausgangspunkt und und das Ziel ist:\\ Ausgangspunkt: $P(S_n = k) = $ \antwortbox[6cm] \vspace{1em} \\ Ziel: $\mathcal{N}(\mu,\sigma ^2) = \mathcal{N}(np)(npq) = $ \antwortbox[6cm] - \item Zuerst müssen wir die Binomialverteilung als diskrete Verteilung in eine stetige Verteilung überführen. Das Aspekt der Binomialverteilung welcher die Diskretheit forciert, ist die Fakultät. Die Fakultät der Reelen Zahlen ist die so genannte Gamma ($\Gamma$) Funktion. Zeichne in dem folgenden Koordinatensystem, in dem auch die Gammafunktion abgebildet ist, die Funktionswerte der Fakultät (welche nur auf $\mathbb{N}$ definiert ist) für $x\in \{0,1,2,3,4,5\}$ ein: + \item Wir wollen die Binomialverteilung als diskrete Verteilung in eine stetige Verteilung überführen. Das Aspekt der Binomialverteilung welcher die Diskretheit forciert, ist die Fakultät. Die Fakultät der Reelen Zahlen ist die so genannte Gamma ($\Gamma$) Funktion. \\ + + Zeichne in dem folgenden Koordinatensystem, in dem auch die Gammafunktion abgebildet ist, die Funktionswerte der Fakultät (welche nur auf $\mathbb{N}$ definiert ist) für $x\in \{0,1,2,3,4,5\}$ ein: \begin{figure}[H] \centering \begin{tikzpicture} \begin{axis}[ - domain=0.1:6, + domain=0.1:4.5, samples=300, axis lines=middle, xlabel={$x$}, - ylabel={$\Gamma(x)$}, + ylabel={$y$}, xmin=0, xmax=6, ymin=0, - ymax=10, - grid=both + ymax=26, + grid=both, + legend pos=north east ] \addplot[ blue, @@ -369,10 +374,16 @@ \[ P(S_n=k)= \frac{1}{\sqrt{2\pi npq}}\left(\frac{k}{np}\right)^{-k} \left(\frac{n-k}{nq}\right)^{-n+k} \Leftrightarrow e^{ln(P(S_n = k))} \sim e^{ln\left(\frac{1}{\sqrt{2\pi npq}}\left(\frac{k}{np}\right)^{-k} \left(\frac{n-k}{nq}\right)^{-n+k}\right)} \] + Beschreibe kurz, was genau die obere Gleichung aussagt und wieso wir einfach die $e$-Funktion und den (natürlichen)Logarithmus anwenden können! + \answer{3} + Im Folgenden betrachten wir erst mal nur den Ausdruck im $ln()$ welchen wir mithilfe der Logarithmengesetze umformen können: \[ \ln\left( \left(\frac{k}{np}\right)^{-k} \left(\frac{n-k}{nq}\right)^{-n+k} \right) = (-np - z\sqrt{npq}) \ln\left(1+z\sqrt{\frac{q}{np}}\right) + (-nq + z\sqrt{npq}) \ln\left(1-z\sqrt{\frac{p}{nq}}\right) \] + Nenne die Rechengesetze und Substitutionen, welche vermutlich für diese Umformung genutzt wurden! + \answer{3} + Diese Darstellung des Thermes sieht zwar erst mal größer aus, hat aber den Vorteil, dass er aus zwei sich sehr ähnlich aussehenden Teilen besteht. Das machen wir uns zu Nutzen! Mithilfe der Tailor-Approximation für den Logarithmus lassen sich nämlich folgende Abschätzungen beweisen: \[ ln(1-\alpha) \sim -\alpha-\frac{\alpha^2}{2} @@ -397,6 +408,5 @@ \end{align*} Daher konvergiert eine Binomialverteilte Zufallsvariable (für $n\rightarrow\infty$) gegen die Dichtefunktion der Normalverteilung. \end{enumerate} - \section{Platz für Notizen und Nebenrechnungen:} \label{LastPage} \end{document} \ No newline at end of file diff --git a/material/ab3_Lös.pdf b/material/ab3_Lös.pdf new file mode 100644 index 0000000000000000000000000000000000000000..3ecbdd7764113d7cf54127f9350ecadfbcf16762 GIT binary patch literal 342493 zcmag_1yqz>+c%7>s0hL!Dm|2xgdjb%Iw&2|jl|I1jiN9jNJH}y z!TWxn=l!qmTYuK_ysnwqaqMFsef&;3dC@n|*jc%;={g4M1{z0t{xo58Q9Y-!);Gfz z6r^HPq+%1ZRLnO!lCX$`~|BXud|9b}q^hY=JFA=gJX`9IVDXCdSNdj|jiaD|=Y|D7$G zRP3DK7X>P|H>M6$+W+hRIG+En8E;H291wP3kcFNDLKIJD5;$a`N(l zc~t-Yu<27$y^^`M(k)wE3(n50)5i{nv&$vZ*$k8J z)n4vLrtU^`S-9;Lc+Mi1f#lxJ)6M~p`TZ8XO;?bO?+KXPJgyv1W;Z)G&q*5e-o7!J 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before title={\refstepcounter{commoncounter}}, % KRITISCH: vor dem Titel! + title={Zusatzaufgabe: #1}, + colback=white, + colframe=white!75!red, + fonttitle=\bfseries, + boxrule=0.6mm, + coltitle=black, + rounded corners, + before skip=10pt, + after skip=10pt +} + +\newtcolorbox{merkbox}[1][]{ + title={Merken: #1}, + colback=white, + colframe=white!75!red, + fonttitle=\bfseries, + boxrule=0.6mm, + coltitle=black, + rounded corners, + before skip=10pt, + after skip=10pt +} + +\definecolor{SatzColor}{RGB}{66,133,244} +\newtcolorbox{satzbox}[1][]{ + before title={\refstepcounter{commoncounter}}, % KRITISCH: vor dem Titel! + title={Satz \thecommoncounter: #1}, + colback=SatzColor!5, + colframe=white!75!SatzColor, + fonttitle=\bfseries, + boxrule=0.6mm, + coltitle=black, + rounded corners, + before skip=10pt, + after skip=10pt +} + +\newcommand{\antwortbox}[1][4cm]{% + \raisebox{-0.4\height}{\fbox{\rule{0pt}{3em}\hspace{#1}}}% +} + +\definecolor{codegreen}{rgb}{0,0.6,0} +\definecolor{codeblue}{rgb}{0,0,0.8} +\definecolor{codered}{rgb}{0.8,0,0} +\definecolor{lightgray}{rgb}{0.95,0.95,0.95} + +\lstdefinestyle{CStyle}{ + language=C, + basicstyle=\ttfamily\small, % Monospace-Schrift + keywordstyle=\color{blue}\bfseries, % Schlüsselwörter fett und blau + stringstyle=\color{red}, % Strings rot + commentstyle=\color{codegreen}, % Kommentare grün + backgroundcolor=\color{lightgray}, % Hintergrundfarbe + numbers=left, % Zeilennummern links + numbersep=10px, % Abstand zwischen Zeilennummern und Code + numberstyle=\color{gray}\texttt, + stepnumber=1, % Zeilennummerierung Schrittweite 1 + frame=single, % Rahmen um den Code + tabsize=4, % Tabulatorgröße + breaklines=true, % Zeilenumbruch aktivieren + captionpos=below, + showstringspaces=false, + xleftmargin=15pt, % Linker Rand für den Code (verschiebt alles nach rechts) +} + + +\lstdefinestyle{ShellStyle}{ + language=Bash, + basicstyle=\ttfamily\small, % Monospace-Schrift + keywordstyle=\color{black}, % Schlüsselwörter fett und blau + stringstyle=\color{black}, % Strings rot + commentstyle=\color{codegreen}, % Kommentare grün + backgroundcolor=\color{lightgray}, % Hintergrundfarbe + numbers=left, % Zeilennummern links + numbersep=10px, % Abstand zwischen Zeilennummern und Code + numberstyle=\color{gray}\texttt, + stepnumber=1, % Zeilennummerierung Schrittweite 1 + frame=single, % Rahmen um den Code + tabsize=4, % Tabulatorgröße + breaklines=true, % Zeilenumbruch aktivieren + captionpos=below, + showstringspaces=false, + xleftmargin=15pt, % Linker Rand für den Code (verschiebt alles nach rechts) +} + + +% Umgebung für Listings mit Titel und Zähler +\newenvironment{codeexample}[1][]{ + \refstepcounter{commoncounter} % Zähler erhöhen + \lstset{ + style=CSharpStyle, + caption={Listing \thecommoncounter: #1}, % Titel mit Zähler + label={listing:\thecommoncounter} + } +}{} + +\makeatletter +% Internen Zähler von subsubsection auf commoncounter umbiegen +\let\c@subsubsection\c@commoncounter + +% Anzeige der Nummer angleichen +\renewcommand{\thesubsubsection}{\thecommoncounter} +\makeatother + +\setlength{\parindent}{0pt} + +\sethlcolor{black!10} % Farbe des "Textmarkers" + +\newcommand{\curdate}{09.07.2026} +\newcommand{\curtitle}{ZSG Movire-Laplace} + +\title{\curtitle} +\author{Fierke, Janik, Weidlich} +\pagestyle{scrheadings} +\date{\curdate} +\begin{document} + \ihead{\normalfont\normalsize Mathematik} + \chead{\normalfont\normalsize \curtitle} + \ohead{\normalfont\normalsize \curdate} + \ofoot{\normalfont\normalsize Seite {\pagemark} von \pageref{LastPage}} + \ifoot{\normalfont\normalsize Fierke, Janik, Weidlich} + + \section{\curtitle} + Das Galtonbrett lässt sich wir folgt formal mathematisch formulieren: + \begin{definitionbox}[Modell des Galtonbretts] + Sei $(\Omega, \mathcal{F}, P)$ ein Wahrscheinlichkeitsraum. \\ + Der Fall einer Murmel durch ein Galtonbrett mit \hl{$\bm{n \in \mathbb{N}}$} Reihen wird modelliert durch eine Folge von stochastisch unabhängigen und identisch verteilten Zufallsvariablen $X_1, X_2, \dots, X_n$, wobei \hl{$\bm{X_i \in \{0, 1\}}$}. + Dabei beschreibt \hl{$\bm{X_i = 1}$} den Fall nach rechts in der $i$-ten Reihe und \hl{$\bm{X_i = 0}$} 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 $p = q = 0.5$.\\ + Die Endposition der Murmel im Fach $k \in \{0, 1, \dots, n\}$ wird durch die Summe der Rechtsabbiegungen beschrieben. Wir definieren die Zufallsvariable: \[\colorbox{black!10}{$\displaystyle \bm{S_n=\sum_{i=1}^n X_i}$}\] Hierbei ist $S_n$ Binomialverteilt. + \end{definitionbox} + + \textbf{\underline{Aufgabe 1:}} Um die Definition genauer zu verstehten, ordne die in der Definition hervorgehobenen Teile den Folgenden Abbildungen des Galtonbrettes zu: + \begin{figure}[H] + \centering + % Erstes Bild + \begin{minipage}[t]{0.45\textwidth} + \centering + \includegraphics[width=\linewidth]{../images/AB3_Galton_L.png} + \caption{Darstellung des Galtonbretts nach Brünner (2025)} + \label{fig:galton} + \end{minipage} + \hfill + % Zweites Bild + \begin{minipage}[t]{0.45\textwidth} + \centering + \includegraphics[width=\linewidth]{../images/AB3_X_L.png} + \caption{Darstellung des Falls einer Murmel im Galtonbrett.} + \label{fig:galtonXi} + \end{minipage} + \end{figure} + \textbf{\underline{Aufgabe 2:}} Sobald sehr viele Murmeln durch das Galtonbrett gewandert sind, entsteht eine Verteilung, wie in Grafik 3 für ein symmetrisches Brett dargestellt wird. + \begin{enumerate} + \item Probiere das ganze selbst einmal auf folgender Website aus: \url{https://www.arndt-bruenner.de/mathe/scripts/galtonbrett.htm}.\\ Was fällt dir auf? \\ \textit{Achtung: stelle p = 0,5 für ein symmetrisches Brett ein!}\\ + Die Verteilung entspricht einer Binomialverteilung. + \item Recherchiere selbstständig und zeichne den Graphen der Normalverteilung über die Verteilung des Galtonbrettes ein: + \begin{figure}[H] + \centering + \includegraphics[width=0.8\linewidth]{../images/AB3_GaltonVersuche_L.png} + \caption{Galtonbrett nach c.a. 7000 durchgefallenen Murmeln nach Brünner (2025)} + \end{figure} + \item Wiederhole oder Recherchiere selbstständig die Eigenschaften der Normalverteilung und Vergleiche diese mit der Binomialverteilung (die Verteilung des Galtonbrettes): + \begin{table}[H] + \centering + \begin{tabularx}{\textwidth}{|X|X|} + \hline + \centering\textbf{Normalverteilung} & + \centering\arraybackslash\textbf{Binomialverteilung} \\ + \hline + Stetig & Diskret \\ + Parameter $\mu, \sigma$ & Parameter n, p\\ + Glockenkurve & Histogramm \\ + $P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$ & $f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}$\\ + \hline + \end{tabularx} + \end{table} + \end{enumerate} + \vspace{4em} + Wir wollen nun das schöne mathematische Phänomen, welches sich in Aufgabe 2.2 offenbart, genauer betrachten. Hierzu formulieren wir dies zuerst mathematisch formal sauber: + \begin{satzbox}[Zentraler Grenzwertsatz von de Movire-Laplace] + 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 $q = 1-p$ die Gegenwahrscheinlichkeit. (Beispielsweise bei einem Galtonbrett)\\ + Für große $n$ lässt sich die Wahrscheinlichkeit, dass $S_n$ den Wert $k \in \{x| x \in\mathbb{N}_0 \land x \leq n\}$ annimmt, durch die Dichtefunktion der Normalverteilung annähern: + \[ + P(S_n = k) \sim \mathcal{N}(\mu,\sigma ^2) + \] + wobei $\mathbb{E}(X)=\mu$, $Var(X)=\sigma ^2$ und $\frac{k-np}{\sqrt{npq}}$ begrenzt ist. + \end{satzbox} + Diesen Satz wollen wir nun Semi-Formal beweisen.\\ + \textbf{\underline{Aufgabe 3:}} Fülle die Lücken im Beweis aus. Wenn du dir unsicher bist kannst du dir alle fehlenden Textbausteine auf dem Zusatzblatt ansehen. + \begin{enumerate} + \item Bevor wir den Satz beweisen, ist es wichtig, die Aussage zu verstehen. Formuliere die Kernaussage den Satzes kurz in eigenen Worten:\\ + Die Binomialverteilung lässt sich mithilfe der Normalverteilung für eine hohe Anzahl an Durchführungen approximieren. + \vspace{3em} + \item Zuerst muss uns nochmal klar werden, was unser Ausgangspunkt und und das Ziel ist: + \vspace{3em} + Ausgangspunkt: $P(S_n = k) = \frac{n!}{k!(n-k)!} p^k q^{n-k}$ \vspace{2em} \\ + Ziel: $\mathcal{N}(\mu,\sigma ^2) = \mathcal{N}(np)(npq) = \frac{1}{\sqrt{2\pi npq}}e^{\left(-\frac{(k-np)^2}{2npq}\right)}$ \vspace{3em} + \item Wir wollen die Binomialverteilung als diskrete Verteilung in eine stetige Verteilung überführen. Das Aspekt der Binomialverteilung welcher die Diskretheit forciert, ist die Fakultät. Die Fakultät der Reelen Zahlen ist die so genannte Gamma ($\Gamma$) Funktion. \\ + + Zeichne in dem folgenden Koordinatensystem, in dem auch die Gammafunktion abgebildet ist, die Funktionswerte der Fakultät (welche nur auf $\mathbb{N}$ definiert ist) für $x\in \{0,1,2,3,4\}$ ein: + \begin{figure}[H] + \centering + \begin{tikzpicture} + \begin{axis}[ + domain=0.1:4.5, + samples=300, + axis lines=middle, + xlabel={$x$}, + ylabel={$y$}, + xmin=0, + xmax=5, + ymin=0, + ymax=26, + grid=both, + legend pos=north west + ] + + \addplot[ + blue, + thick + ] + {sqrt(2*pi)*((x+1/2)^(x+1/2))*exp(-(x+1/2))}; + \addlegendentry{Fortsetzung} + + \addplot[ + only marks, + mark=*, + mark options={fill=red, draw=red, scale=1.2} + ] + coordinates { + (0, 1) + (1, 1) + (2, 2) + (3, 6) + (4, 24) + }; + \addlegendentry{Fakultät $x!$} + + \end{axis} + \end{tikzpicture} + \end{figure} + Mithilfe dieser Funktion lässt sich die Fakultät leicht approximieren. Diese Approximation nennt sich Sterling Approximation und besagt: + \[ + n! \sim \sqrt{2\pi n} \left(\frac{n}{e}\right)^n \sim \Gamma(n) + \] + Setzen wir das in die Gleichung für $P(S_n = k)$ ein, erhalten wir nach umstellen: + \[ + P(S_n = k) \sim \frac{1}{\sqrt{2\pi}}\sqrt{\frac{n}{k(n-k)}} \left(\frac{np}{k}\right)^k \left(\frac{nq}{n-k}\right)^{n-k} + \] + \item Aus dieser Formel lässt sich nun leicht schon einmal der Vorfaktor der Normalverteilung erschließen. Hierzu und zu Vereinfachung weiterer Rechnungen standardisieren wir $S_n$. Dies entspricht auch einer einfachen Substitution und daher vollkommen legitim. Hierzu definieren wir: + \[ + Z_n = Z_n=\frac{S_n-E(S_n)}{\sigma(S_n)} + \] + nimmt $Z_n$ die folgenden Werte an: + \[ + z= \frac{k-np}{\sqrt{npq}} + \] + \vspace{3em} + Nach dem Umstellen nach $k$ und $n-k$ folgt dann: + \[ + k= np + z\sqrt{npq} + \] + \vspace{3em} + \[ + n-k= nq - z\sqrt{npq} + \] + \vspace{3em} + Multiplizieren wir nun diese beiden Ausdrücke miteinander erhalten wir: + \[ + k(n-k)= n^2pq + (nz\sqrt{nqp}(q-p)-z^2npq) + \] + Da wir uns nur für eine große Anzahl an Durchführungen also große $n$ ($n\rightarrow \infty$) interessieren, ist beobachtbar, dass für große n der lineare Teil kaum noch einen Einfluss auf den Term hat. Daher kann dieser in diesem Fall weggelassen werden. Daher gilt: + \[ + k(n-k) \sim n^2pq + \] + Setzen wir dies nun in unsere diskrete Variante von $P(S_n = k)$ ein, erhalten wir den passenden Vorfaktor: + \begin{align*} + P(S_n = k) &\sim \frac{1}{\sqrt{2\pi}}\sqrt{\frac{n}{k(n-k)}} \left(\frac{np}{k}\right)^k \left(\frac{nq}{n-k}\right)^{n-k} \\ + &\sim \frac{1}{\sqrt{2\pi}}\sqrt{\frac{n}{n^2pq}} \left(\frac{np}{k}\right)^k \left(\frac{nq}{n-k}\right)^{n-k} \vspace{1em} \\ + &= \frac{1}{\sqrt{2\pi}}\frac{1}{\sqrt{npq}} \left(\frac{np}{k}\right)^k \left(\frac{nq}{n-k}\right)^{n-k} \\ + &= \frac{1}{\sqrt{2\pi npq}}\left(\frac{k}{np}\right)^{-k} \left(\frac{n-k}{nq}\right)^{-n+k} + \end{align*} + \item Zum Schluss müssen wir noch die $e$-Funktion mit in die Gleichung bekommen. Das können wir machen, indem wir die Identität ausnutzen: $x = e^{ln(x)}$. Also gilt insbesondere, dass: + \[ + P(S_n=k)= \frac{1}{\sqrt{2\pi npq}}\left(\frac{k}{np}\right)^{-k} \left(\frac{n-k}{nq}\right)^{-n+k} \Leftrightarrow e^{ln(P(S_n = k))} \sim e^{ln\left(\frac{1}{\sqrt{2\pi npq}}\left(\frac{k}{np}\right)^{-k} \left(\frac{n-k}{nq}\right)^{-n+k}\right)} + \] + Beschreibe kurz, was genau die obere Gleichung aussagt und wieso wir einfach die $e$-Funktion und den (natürlichen)Logarithmus anwenden können!\\ + Da $e(ln(x))=x$ können wir diese Identität auf beide Seiten der Gleichung anwenden, ohne das Ergebnis zu verändern. Der Logarithmus ist nur für $x>0$ definiert, das bereitet hier aber keine Probleme, da $0 Date: Sun, 12 Jul 2026 17:33:59 +0200 Subject: [PATCH 3/5] Zeitplan --- content/Arbeitsmaterialien.tex | 2 +- content/Zeitplan.tex | 4 +--- galton.pdf | Bin 479415 -> 482269 bytes 3 files changed, 2 insertions(+), 4 deletions(-) diff --git a/content/Arbeitsmaterialien.tex b/content/Arbeitsmaterialien.tex index b1dba69..4b0aede 100644 --- a/content/Arbeitsmaterialien.tex +++ b/content/Arbeitsmaterialien.tex @@ -33,7 +33,7 @@ Bitte bedenken Sie, zumindest Lösungsskizzen zu den Aufgaben zu geben! Besser n % ln und umformen (Klassenstufe! -> Anschätzung und Ergebnis vorgeben, nicht rechnen) % Rücksubstitution machen Das dritte Arbeitsblatt dient dem Beweis des zentralen Grenzwertsatzes von Movire-Laplace. Ziel ist es hierbei explizit nicht, dass die Schüler*innen den Beweis in jedem Detail, wie im Teil "Mathematischer Hintergrund", verstehen aber die Grundlegenden Argumentationen und mathematischen Tricks nachvollziehen. Außerdem ist vor allem das Beweisen etwas, dass im Rahmenlehrplan (\cite{lisum-2022} und \cite{lisum-2023}) nicht gefordert ist, wodurch es sich sehr gut für eine MatheAG oder Zusatzstunde handelt, in der Inhalte über den klassischen Unterricht hinaus behandelt werden.\\ -Für das Arbeitsblatt sollten die Schüler*innen bereits mit der Binomialverteilung, Zufallsvariablen, Logarithmen sowie der e-Funktion vertraut sein und mit etwas Formalismus umgehen können. Tiefgreifende Kenntnisse zur Normalverteilung sind nicht vorausgesetzt und werden auf dem Arbeitsblatt in ausreichender Form wiederholt bzw. können selbständig aufbereitet werden. Es ist empfohlen mindestens eines der Arbeitsblätter Galton x Pascall oder Galton x Plinko vorher bearbeitet zu haben, um das Konzept des Galtonbrettes und dessen Zusammenhang mit der Binomialverteilung verstanden zu haben.\\ +Für das Arbeitsblatt sollten die Schüler*innen bereits mit der Binomialverteilung, Zufallsvariablen, Logarithmen sowie der e-Funktion vertraut sein und mit etwas Formalismus umgehen können. Tiefgreifende Kenntnisse zur Normalverteilung sind nicht vorausgesetzt und werden auf dem Arbeitsblatt in ausreichender Form wiederholt bzw. können selbständig aufbereitet werden. Es ist empfohlen mindestens das Arbeitsblätt \emph{Pascalsches Dreieck} und optimaler Weise auch noch das Arbeitsblatt \emph{Galton x Plinko} vorher bearbeitet zu haben, um das Konzept des Galtonbrettes und dessen Zusammenhang mit der Binomialverteilung verstanden zu haben.\\ Auf dem Arbeitsblatt wird die formale Modellierung des Galtonbrettes vorgestellt und soll durch eine Zuordnungsaufgabe durch die Schüler*innen vertieft werden. Im Anschluss soll der Bezug zwischen Binomial- und Normalverteilung wiederholt bzw. erarbeitet werden. Das Kernstück des Arbeitsblattes stellt der Lückentext-Ähnliche Beweis dar. Hier sollen die Schüler einige Mathematische Argumentationen nachvollziehen und einige Umformungen selbst durchführen, um den Beweis nachvollziehen zu können. Sollten die Schüler leistungsschwächer sein oder an einem Punkt fest Stecken gibt es zur Differenzierung Lösungskarten für die Lücken, welche den Schülern in nicht sortierter Reihenfolge zur Verfügung gestellt werden können. %Noch Checkfragen in den Beweis einbauen? \ No newline at end of file diff --git a/content/Zeitplan.tex b/content/Zeitplan.tex index 4e2b38f..3208d92 100644 --- a/content/Zeitplan.tex +++ b/content/Zeitplan.tex @@ -1,3 +1 @@ -Hier ist kein minutengenauer Zeitplan nötig, aber eine grobe Übersicht über den Zeitplan ist sinnvoll. Insbesondere, wie viel Zeit Sie für die einzelnen Themen bzw. Arbeitsmaterialien veranschlagen. - -Das ganze muss nicht in eine Doppelstunde passen, aber es ist sinnvoll, wenn Sie die normalen Doppelstunden als grobe Richtlinie verwenden. Der Zeitplan kann z.B. gerne als eine Tabelle dargestellt werden. \ No newline at end of file +Um eine tiefgreifende Auseinandersetzung mit dem Galatonbrett zu ermöglichen empfehlen wir alle drei Arbeitsblätter anzubieten. Wir empfehlen mit dem Arbeitsblatt \emph{Pascalsches Dreieck} (c.a. 25 min) zu beginnen, daran das Arbeitsblatt \emph{Galton x Plinko} (c.a. 25 min) anzuschließen und das Arbeitsblatt \emph{ZSG Movire-Laplace} (c.a. 40min) zum Abschluss zu benutzen. 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zB*`9Jp#M&{v&zdLlCbAQ5(Gy8rFJgkz{x`+;pp>Uvve?5EHt>XW2p=H$G?a9j|&8+ zKNi~w3&vty=%8&Zok08=D|+bO&r*)YK+)O$4jOfVAk@h^sT5@?FP0%8n~NS8h8%aC4ndt#@2x;0aM4?F@yCS0D_5)H$dJ06=7kp zcrmcVEE55DkwO>5jG6%#7Pj_yq5q*5B#3FTlm-U^OM^`kDVj%#+z#d*;ccx>vU5J#hatp$LbFDqBF&a|Aasfhk z*n<=kAYNI5@o%uz95TSvaC=9p2CB}vkpgjn&i%-^zT+kA%HjzOTu*L`K z!c90P)(ZEEERqhDp%=_xfUVfL860s)ST7cWbu8gcxU2m)Lisz^3PBAN`G4+=;Vucg znEOecsBssAXrR{ia|NJC1NQ@ zflGOeC%l*e4l@zYqo819W6bk^7km^y From cd4de07333a33e83c4a6bc4a0b66608c7d166afb Mon Sep 17 00:00:00 2001 From: Luca Weidlich Date: Mon, 13 Jul 2026 07:59:12 +0200 Subject: [PATCH 4/5] Zeitplan etwas differenzierter betrachtet --- content/Zeitplan.tex | 6 ++++++ galton.pdf | Bin 482269 -> 483740 bytes 2 files changed, 6 insertions(+) diff --git a/content/Zeitplan.tex b/content/Zeitplan.tex index 3208d92..2b4e930 100644 --- a/content/Zeitplan.tex +++ b/content/Zeitplan.tex @@ -1 +1,7 @@ +Da sich das Themengebiet über einige Niveau- und Klassenstufen des Brandenburger Rahmenlehrplanes streckt \cite{lisum-2023}, und das Galtonbrett sich auch sehr gut eignet, auf verschiedenen Komplexitätsstuufen zu unterrichten, schlagen wir folgende Unterteilung vor:\\ + +Ab 10. Klasse:\\ +Eine Grundlegende Auseinandersetzung mit dem Galtonbrett und den Zusammenhang mit der Binomialverteilung und dem Pascallschem Dreieck, lässt sich wunderbar ohne höhere mathematische Inhalte erarbeiten. Hierzu eignen sich insbesondere das Arbeitsblatt \emph{Pascalsches Dreieck} (c.a. 25 min) und \emph{Galton x Plinko} (c.a. 25 min). Wir würden auch empfehlen beide zu bearbeiten. Wenn eine ausblicksartige Auseinandersetzung im Zusammenhang mit der Normalverteilung gewünscht ist (ohne in die mathematischen Details dieser einzutauchen) kann hierzu auch noch Aufgabe 1 bis Aufgabe 3 des Arbeitsblattes \emph{ZSG Movire-Laplace} (c.a. 15min) mit eingebunden werden.\\ + +Ab 12. Klasse:\\ Um eine tiefgreifende Auseinandersetzung mit dem Galatonbrett zu ermöglichen empfehlen wir alle drei Arbeitsblätter anzubieten. Wir empfehlen mit dem Arbeitsblatt \emph{Pascalsches Dreieck} (c.a. 25 min) zu beginnen, daran das Arbeitsblatt \emph{Galton x Plinko} (c.a. 25 min) anzuschließen und das Arbeitsblatt \emph{ZSG Movire-Laplace} (c.a. 40min) zum Abschluss zu benutzen. 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Date: Mon, 13 Jul 2026 08:29:27 +0200 Subject: [PATCH 5/5] =?UTF-8?q?tobi=20=C3=84nderungen=20und=20Sigma?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- content/Einordnung_Lehrplan.tex | 2 +- content/Mathematischer_Hintergrund.tex | 50 ++++++++++++------------- content/Weiterfuehrende_Ideen.tex | 4 +- content/Zeitplan.tex | 2 +- galton.pdf | Bin 483740 -> 483826 bytes 5 files changed, 29 insertions(+), 29 deletions(-) diff --git a/content/Einordnung_Lehrplan.tex b/content/Einordnung_Lehrplan.tex index 0350b90..4be146f 100644 --- a/content/Einordnung_Lehrplan.tex +++ b/content/Einordnung_Lehrplan.tex @@ -1,4 +1,4 @@ Ein erster Kontakt mit der Binomialverteilung tritt in Form von Binomialkoeffizienten in der zehnten Klasse des Gymnasiums als Teil der Niveaustufe H auf (\cite{lisum-2023}). -Vertieft wird die Binomialverteilung dann im zweiten Semester der Oberstufe und wird im vierten Semester durch die Normalverteilung ergänzt (\cite{lisum-2022}). +Vertieft wird die Binomialverteilung dann im zweiten Semester der Oberstufe und wird im vierten Halbjahr durch die Normalverteilung ergänzt (\cite{lisum-2022}). diff --git a/content/Mathematischer_Hintergrund.tex b/content/Mathematischer_Hintergrund.tex index 3309762..739509e 100644 --- a/content/Mathematischer_Hintergrund.tex +++ b/content/Mathematischer_Hintergrund.tex @@ -1,7 +1,7 @@ %"ltex.language": "de-DE" \subsection{Beweis des zentralen Grenzwertsatzes} -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}. +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 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 $X_i \in \{0, 1\}$. @@ -42,19 +42,19 @@ Um jeden Ausgang des Galtonbrettes durchnummeriert von links nach rechts untersc \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} -Um diese summierte Zufallsvariable genauer zu verstehen betrachten wir zuerst den Binomialkoeffizienten. +Um diese summierte Zufallsvariable genauer zu verstehen, betrachten wir zuerst den Binomialkoeffizienten. \begin{definition}[Binomialkoeffizienten]{binomialkoeffizienten}{\cite{kosenkova_stochastik_2025}} Für $n\in\mathbb{N}_0$ und $k\in\{0,\dots,n\}$ definieren wir den \textbf{Binomialkoeffizienten} \[ \binom{n}{k}=\frac{n!}{k!(n-k)!} \] - wobei die Zahl $\binom{n}{k}$ die Anzahl der Möglichkeiten, aus einer $n$-elementigen Menge genau $k$-Elemente auszuwählen, angibt. + 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 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}} +\begin{definition}[Binomialmodell]{binModell}{nach \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 \[ - \Sigma = \{0,1,...,n\}. + A = \{0,1,...,n\}. \] In diesem Modell ist \[ @@ -62,7 +62,7 @@ Betrachten wir nun den Binomialkoeffizienten in Bezug auf $S_n$, beschreibt dies \] eine Zähldichte. \end{definition} -Im Fall des Galtonbrettes können wir $\Sigma = S_n$ wählen. +Im Fall des Galtonbrettes können wir $A = S_n$ wählen. \begin{definition}[Binomialverteilung]{binZähldichte}{\cite{kosenkova_stochastik_2025}} Die Wahrscheinlichkeitsverteilung mit der Zähldichte $Bin_{n,p}(\{k\})$ auf $\{0,...,n\}$ heißt Binomialverteilung zu den Parametern $n, p$. \end{definition} @@ -72,17 +72,17 @@ Anschließend ist folgendes zu bemerken: Daher gilt $S_n \sim \mathcal{B}in_{n,p}$ \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 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: $S_n \sim \mathcal{B}in_{n,p}$. + Nach \cref{bem:XiB} und \cref{def:Sn} ist $S_n$ die Summe von $n$ unabhängigen Bernoulliverteilten Zufallsvariablen $X_i \sim \mathcal{B}_p$. Also stellt $S_n$ die Anzahl der Erfolge von n-Wiederholungen von unabhängigen identisch Bernoulliverteilten Zufallsexperimenten dar. + Daraus folgt, dass das Galtonbrett mit der Zufallsvariable $S_n$ ein Binomialmodell nach \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] - Sei X eine stetige Zufallsvariable mit Dichte + Sei $X$ eine stetige Zufallsvariable mit Dichte \[ f(x)= \frac{1}{\sqrt{2\pi \sigma ^2}}\cdot e^{-\frac{\left( x-\mu \right)^2}{2\sigma ^2}}\] - und Parametern $\mu \in \mathbb{R}, \sigma > 0$, wobei $\mu$ den Erwartungswert und $\sigma ^2$ die Varianz von X darstellt, so nennt man X \emph{Normalverteilt} und schreibt: - \[ x \sim \mathcal{N}(\mu, \sigma ^2)\] + und Parametern $\mu \in \mathbb{R}, \sigma > 0$, wobei $\mu$ den Erwartungswert und $\sigma ^2$ die Varianz von $X$ darstellt, so nennt man $X$ \emph{Normalverteilt} und schreibt: + \[ X \sim \mathcal{N}(\mu, \sigma ^2)\] \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 persé 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}. +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 persé 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}. \begin{satz}[Zentraler Grenzwertsatz von de Moivre-Laplace]{moivrelaplace}{nach \cite{kosenkova_stochastik_2025}} 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 $q = 1-p$. Für große $n$ lässt sich die Wahrscheinlichkeit, dass $S_n$ den Wert $k \in \{x| x \in\mathbb{N}_0 \land x \leq n\}$ annimmt, durch die Dichtefunktion der Normalverteilung annähern: \[ @@ -168,16 +168,16 @@ Für große $n$ wird die direkte Berechnung der Binomialverteilung aufgrund der \item[(\ref{eq5:3})] Gleichheit für $k$ aus der Standardisierung einsetzen: $k = np + z\sqrt{npq}$. \item[(\ref{eq5:4})] Distributivgesetz für $-1$ anwenden und Brüche durch Aufteilen der Addition und Einfügen der multiplikativen Identität $\frac{\sqrt{np}}{\sqrt{np}}$ (selbes für $nq$) vereinfachen. \end{itemize} - Die \cref{def:taylorpolynom} beweist die Taylor-Approximation für den Logarithmus: $ln(1-\alpha) = -\alpha-\frac{\alpha^2}{2}+O(\alpha^3)$ und $ln(1+\alpha) = \alpha-\frac{\alpha^2}{2}+O(\alpha^3)$ für $\alpha \in (-\infty, 1)$. Hierbei steht $O(\alpha^3)$ für die Landau-Notation. + Die \cref{def:taylorpolynom} beweist die Taylor-Approximation für den Logarithmus: $ln(1-\alpha) = -\alpha-\frac{\alpha^2}{2}+O(\alpha^3)$ und $ln(1+\alpha) = \alpha-\frac{\alpha^2}{2}+O(\alpha^3)$ für $\alpha \in (-1, 1)$. Hierbei steht $O(\alpha^3)$ für die Landau-Notation. Wir definieren: \[ \beta := \sqrt{\frac{q}{np}}; \quad \gamma := \sqrt{\frac{p}{nq}} \] - Um diese Approximation zu verwenden, muss sicher sein, dass $-1 < |\alpha| < 1$. Das heißt, dass $\forall z \in \mathbb{R}: \lim_{n\rightarrow\infty} -1 < |z\cdot\beta|<1$. (Für $z\cdot\gamma$ analog) Hierbei genügt die Abschätzung gegen unendlich, da wir die Annäherung nur für große Zahlen beweisen wollen.\\ + Um diese Approximation zu verwenden, muss sicher sein, dass $-1 < \alpha < 1$. Das heißt, dass $\forall z \in \mathbb{R}: \lim_{n\rightarrow\infty} -1 < z\cdot\beta<1$. (Für $z\cdot\gamma$ analog) Hierbei genügt die Abschätzung gegen unendlich, da wir die Annäherung nur für große Zahlen beweisen wollen.\\ Sei also $z\in\mathbb{N}$, dann betrachten wir:\\ \begin{align*} - |z\cdot\beta| \overset{Def.}{=} \left|z \cdot \sqrt{\frac{q}{np}}\right| = |z|\cdot \frac{1}{\sqrt{n}}\cdot\sqrt{\frac{q}{p}} \xrightarrow{n\rightarrow\infty} 0\\ + z\cdot\beta \overset{Def.}{=} z \cdot \sqrt{\frac{q}{np}} = z\cdot \frac{1}{\sqrt{n}}\cdot\sqrt{\frac{q}{p}} \xrightarrow{n\rightarrow\infty} 0\\ und \quad -1 < 0 < 1 \end{align*} Da $p$ und $q$ konstante Parameter sind. @@ -225,7 +225,7 @@ Für große $n$ wird die direkte Berechnung der Binomialverteilung aufgrund der \item[(\ref{eq7:2})] Der Bruch wurde quadriert und zusammengefasst. \item[(\ref{eq7:3})] Definition der Standardnormalverteilung angewendet. \end{itemize} - Daher konvergiert eine Binomialverteilte Zufallsvariable (für $n\rightarrow\infty$) gegen die Dichtefunktion der Normalverteilung. + Daher konvergiert eine binomialverteilte Zufallsvariable (für $n\rightarrow\infty$) gegen die Dichtefunktion der Normalverteilung. \end{proof} \begin{lemma}[Asymptotisches Verschwinden der Taylor-Restterme mit Präfix]{restterme} Seien $p \in (0,1)$, $q = 1-p$ und $z$ Werte einer standardisierten, binomialverteilten Zufallsvariable wobei z begrenzt ist. Für die Restterme der Taylor-Entwicklung des Logarithmus multipliziert mit ihren jeweiligen Vorfaktoren gilt für $n \rightarrow \infty$: @@ -306,11 +306,11 @@ Die Stirling-Approximation ist eine mathematische Näherung zur Berechnung der F \[ d_n := \ln{n!}-\left(n+\frac{1}{2}\right)\ln{n+n} \] - Wir zeigen nun, dass + Wir zeigen nun, folgende Aussage gilt: \[ \lim_{n\rightarrow\infty}d_n =\ln\sqrt{2\pi} \] - Es gilt + Es gilt: \begin{align*} d_n-d_{n+1}=&\ln n!-\ln{(n+1)!}-\left(n+\frac{1}{2}\right)\ln n \\&+\left(n+1+\frac{1}{2}\right)\ln{(n+1)}+n-(n+1)\\ =&\ln\frac{n!}{(n+1)!}+\left(n+\frac{1}{2}\right)\left(\ln(n+1)-\ln (n)\right)+\ln(n+1)-1\\ @@ -503,9 +503,9 @@ Bevor wir die Approximation nachweisen, ist eine formale Definition des Satzes \end{satz} \begin{bemerkung} - Das Lagrange-Restglied ist in den 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 $\xi$. + Das Lagrange-Restglied ist in den 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 $n+1$-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". + Dadurch können wir das Restglied auch auffassen als ein Element von $O(x^{n+1})$. 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} @@ -526,7 +526,7 @@ Nun können wir unsere Approximation von $\ln(1-x)=-x-\frac{x^2}{2}+O(x^3)$ zeig \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 $x_0=0$. + Zudem ist die Funktion $\ln(1-x)$ dreimal differenzierbar. Betrachten wir $x\in(-\infty,1)$ und $x_0=0$. Das zweite Taylor-Polynom lautet \begin{align*} @@ -556,7 +556,7 @@ 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. + 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, welcher durch das Langrange-Restglied beziehungsweise die Landaunotation ausgedrückt wird. 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. \end{bemerkung} Im Plot können wir sehen, dass beide Funktionen um unsere Entwicklungsstelle $x_0=0$ herum sehr ähnlich sind. \begin{figure}[H] @@ -587,16 +587,16 @@ Im Plot können wir sehen, dass beide Funktionen um unsere Entwicklungsstelle $x \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 $\ln(1-x) \sim -x\frac{-x}{2}$ plus ein kleiner Fehler ist. +Würden wir die Entwicklung weiterführen, approximiert die rot-gestrichelte Funktion (unser Taylorpolynom) die blaue Funktion $(\ln(1-x))$ immer weiter. Graphisch lässt sich erkennen, dass für Werte im Intervall $I(-0.5,0.5)$ die Funktion $\ln(1-x) \sim -x\frac{-x}{2}$ plus ein kleiner Fehler ist. \subsection{Galtonbrett} 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 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 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 Galton das Brett am 17.02.1874 bei einer Vorlesung in der Royal Society vorgestellt \cite{kit-2022}. 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 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 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/content/Weiterfuehrende_Ideen.tex b/content/Weiterfuehrende_Ideen.tex index 2802f66..c4f990f 100644 --- a/content/Weiterfuehrende_Ideen.tex +++ b/content/Weiterfuehrende_Ideen.tex @@ -1,9 +1,9 @@ -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 wie praktische Erweiterungsmöglichkeiten, die weit über das Standardmodell hinausgehen. +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. \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. Unter bestimmten Voraussetzungen greif hier dann der Martingal-Zentralgrenzwertsatz, wodurch wieder die Normalverteilung angenähert wird \cite{posedel_martingale_2025}. \paragraph{Identisch Verteilt} -Das klassische Brett nutzt auf jeder Ebene die identische Wahrscheinlichkeit $p = 0.5$. Da auch dies nicht unbedingt immer Gewehrleitbar ist, lässt sich diese Bedingung ebenso aufweichen. Unter der Bedingung, dass keine einzelne Reihe die Gesamtvarianz dominiert (Lindeberg-Bedingung), führt der verallgemeinerte zentrale Grenzwertsatz dennoch zu einer Normalverteilung. Die Glockenkurve bleibt erhalten, auch wenn die Elementarschritte auf dem Weg nach unten variieren \cite{posedel_martingale_2025}. +Das klassische Brett nutzt auf jeder Ebene die identische Wahrscheinlichkeit $p = 0.5$. Da auch dies nicht unbedingt immer gewährleistet ist, lässt sich diese Bedingung ebenso aufweichen. Unter der Bedingung, dass keine einzelne Reihe die Gesamtvarianz dominiert (Lindeberg-Bedingung), führt der verallgemeinerte zentrale Grenzwertsatz dennoch zu einer Normalverteilung. Die Glockenkurve bleibt erhalten, auch wenn die Elementarschritte auf dem Weg nach unten variieren \cite{posedel_martingale_2025}. \paragraph{Erweiterung der Mechanik in der Schule: Das schiefe Galtonbrett} Eine physisch leicht umzusetzende Variation ist das Neigen des gesamten Brettes. Durch die Schräglage wird die Wahrscheinlichkeit für einen Fall nach rechts oder links asymmetrisch, sodass $p \neq 0.5$ gilt. Die Pfade der Kugeln folgen nun einer asymmetrischen Binomialverteilung. Für eine große Anzahl an Stiftreihen $n$ nähert sich die Form (nach dem Satz von Moivre-Laplace) zwar immer noch einer Normalverteilung an, jedoch verschieben sich der Erwartungswert auf $\mu = np$ und die Varianz auf $\sigma^2 = np(1-p)$. Die Glockenkurve wandert entsprechend der Neigung aus dem physikalischen Zentrum des Brettes heraus, behält aber ihre charakteristische mathematische Struktur bei, was im Unterricht hervorragend zur Einführung von Parametervariationen genutzt werden kann \cite{schreiter_echt_2022}. \ No newline at end of file diff --git a/content/Zeitplan.tex b/content/Zeitplan.tex index 2b4e930..499904b 100644 --- a/content/Zeitplan.tex +++ b/content/Zeitplan.tex @@ -4,4 +4,4 @@ Ab 10. Klasse:\\ Eine Grundlegende Auseinandersetzung mit dem Galtonbrett und den Zusammenhang mit der Binomialverteilung und dem Pascallschem Dreieck, lässt sich wunderbar ohne höhere mathematische Inhalte erarbeiten. Hierzu eignen sich insbesondere das Arbeitsblatt \emph{Pascalsches Dreieck} (c.a. 25 min) und \emph{Galton x Plinko} (c.a. 25 min). Wir würden auch empfehlen beide zu bearbeiten. Wenn eine ausblicksartige Auseinandersetzung im Zusammenhang mit der Normalverteilung gewünscht ist (ohne in die mathematischen Details dieser einzutauchen) kann hierzu auch noch Aufgabe 1 bis Aufgabe 3 des Arbeitsblattes \emph{ZSG Movire-Laplace} (c.a. 15min) mit eingebunden werden.\\ Ab 12. Klasse:\\ -Um eine tiefgreifende Auseinandersetzung mit dem Galatonbrett zu ermöglichen empfehlen wir alle drei Arbeitsblätter anzubieten. Wir empfehlen mit dem Arbeitsblatt \emph{Pascalsches Dreieck} (c.a. 25 min) zu beginnen, daran das Arbeitsblatt \emph{Galton x Plinko} (c.a. 25 min) anzuschließen und das Arbeitsblatt \emph{ZSG Movire-Laplace} (c.a. 40min) zum Abschluss zu benutzen. Das Arbeitsblatt \emph{Galton x Plinko} ist hierbei als optional zu betrachten und dient lediglich der vertiefenden Auseinandersetzung mit dem Galtonbrett und der Motivation. \ No newline at end of file +Um eine tiefgreifende Auseinandersetzung mit dem Galtonbrett zu ermöglichen empfehlen wir alle drei Arbeitsblätter anzubieten. Wir empfehlen mit dem Arbeitsblatt \emph{Pascalsches Dreieck} (c.a. 25 min) zu beginnen, daran das Arbeitsblatt \emph{Galton x Plinko} (c.a. 25 min) anzuschließen und das Arbeitsblatt \emph{ZSG Movire-Laplace} (c.a. 40min) zum Abschluss zu benutzen. Das Arbeitsblatt \emph{Galton x Plinko} ist hierbei als optional zu betrachten und dient lediglich der vertiefenden Auseinandersetzung mit dem Galtonbrett und der Motivation. \ No newline at end of file diff --git a/galton.pdf b/galton.pdf index f5eda4694b54f71b5f03f3e6aa2525b31bf90186..c6550de6932ade04d9526114a7bde3bdbe211685 100644 GIT binary patch delta 114879 zcmV)SK(fD_!yWR&9k9;{0W+78SOXQap3$pk=3|`s#i)clwX*#-KQmF}$GtwtkB%xC z29LoFe$4YuhYZ=H25c!H$d{lNsA#T`7pu@mN-2_JVWrl8Jek147DA1`XII>;286+Kd^-M;%X4Xy80dciN?nDCWtL36o)3t z(aaS+K?#JR@QM0?$88m3jf%;xB;i#we*|F^rVebMtvN4r)=#of#Z~TO@Atqc)Z5-b z7zZO_qP5%{VG(FO(QY?_)^qk{p2AD@_dd@}KoMK;mi4v7bEKB<+`4Ntygxd<%5R0~ zTB}6Jfz-=d*~aU3gKSmu?0~H^NkWqoql@DKRZ#*s%AO;#q(Y-Y#9f7s8t z2Q_Zic7rqSZsU|zQWR^wMcNB&YZ@s*r0qOY+d;%fOrdMnRR+sOf@bR|2nNgr)~?Pl zs?+3TgY{aFrRI1wRby!FV6D1R5WSL?&dtS=xRj8p0~~P1qvD2h2WwK38;NSQnfJvt zI94o@JJg8|SGr_ylKtb>HE*kTe;U-c=(Lih_rqUsrEh(~hIiyM#y#>cTJ(FDras@Y zx-wiw1D+dP=h3^KI?U2VDsw$AUg2k`4H0{-(oW;rx#K(PK-uv!63B+#yDFe|mGMt& zw~=NJ)#AcE52-7rtJOx1LSvn2UB0>PFu>9=H#ABiyqBi}(Waq$dA-Kge^_0O|I_pP z1F2(Jx;tcDU4xhe(ATtfXc|Z=dXPEl-6C|Luq2@lzC_>~;ZY_K*`1};_2rKt>HcAa zd$g2FdNgTg%N1+K7K93~M0!)4RpO+!4-_)tVE^gPK`6U?u0Rb?Kb z1MV_QYHM^RyVfoj;)1Spc3fHCy0eJ5=Sf_822XH@&ib@VhDe~CfA#Xl&(A)g-~XZ{ zW7%gYD=ZHyyz7ytCKu#>iC%1+&Pfb6#3)KX)=79V<)`t069gEUg3^2&2ywUf)s_DG zCM*KSm8%hgD!XJwbnj}+J627uPz0|HBrPrl`9vF~D*)9SrKQ@urwacx@M~MUg6B@J ze-(jy2d9Qb=JZIbe}|>UEQb{k{XIUFBCG=%AwQob9m(bBOsZTOg&CjEPnJt=JH8|y zl-DCIP8T@$S7>S?%%#qpCv6mNnX5WqJ64AF&ecZOh}7C2tLxF}@*5ThS&dei=Ol4s zTFC^niX}a&K6>gUsaIxK_|pY@mr9P+Zj;KMJGbOZp?_rte{x*nE9A78r;V)XqV05| z>g?;%8m(xCWV9y%I>~SAis{QXNn&w24{`YVH2^6ScL6fRq=Q>kDPb3!vd|uy@~XWa z-j`<%Z3137&53);LH2eo+rzq|i<_`QXEkmewjb_=&S?WyBr{INPQ6_`y^TPJ9PHt| zcJ;b>eW0TGe+peY%}YQ#M!b0EjC%f>*wk6ce+4V_UBXTe(BXYOHy(LHL2xdP#Kx5X z1`j7t+}zFQkmRNw7>(4Zu@(rLNy7YK z_F=bZy)rs(zkGT6{YklGO{xpT$2vvZ`oK5eeKFhlx2GS<4>y8CtR3kOC1HZ?6bmAD zwox1(>r@d%*Z)eHIj&AcjHuVg0 z0hZhaQ|EJ!A_X*XNd@2M@l8=061J^-+2~7jM3Li>I%9IL2$@&4j%OQO8uzYjUF1=M z-@)tyG{*zDb1M3ULFP|+m^L` z!`og>JCE=sL&RC;+1Hi>lAforA?e+!EhQbH^UM&d6~pA6@0UZl_Felr;RPq_vhua1 zG2SuTG{Mt<);{WX2>p6r)2w$FW$OGt9k7zs8`Et+}UVVW3I)Uvp18ov=5 z9e^a+344zrISHbzO4<|35xtjYQ%7*cP}w^|x2I#rUm_h0yYP|_DSAx`a|AIlv7&>RLC6u3fioUgvkqeWIm zFHHG=Rd+*431)c#Et8XyAnxCdoYep$K_R{WeD%W#M*Phw4SU7GVe#hl<0&@7e*}X+ zE|~{NfDjBC%OCyz{_`o1)<-aJ=`d$9&RfJ1fq$gm-gd|(H_U)nYdjM{AEE%~LvWvQ zm~$KFEu#9bp0_=)Yk0uCz$2pUMGDQ_1^UxK9_UfDOKYsZ2`@hDq zKmIGzC79znsMF{22>zk<+WFHbY%t793#-kUvr0*5u}t~kM1^fF?@w2n2d1P7fi%Nd zVC9&!#tPD`!Ci-)l}11;v<*p#qqgRP^b>7F+8UNfdm|-r5qMEx<71j%hT@gi?HRj& z0@UA5S05-h(|F)RuW~d5D}B$^($sSn8(}Xy8+~>N`jj!LMFpi5;V!le+{NZkx!DP$ zENPCs01ajh!9v?Q?$&0?-PV|CYgmwe%HiD7aXB}AicNV0Y-)V!+qb!%gdK#$rJ&RM zBRhCkEzUKDfXV4#PcUg89=|`!DCatV%u$W?b@3n44pGl*P%fw)VoCxGVpR^FDeiq- zH)#V<9GC%#La6cxizxw8fW?*A_9?bG=A<#9g28YhXdBwNqzO@{(XbUUeWHzpm^OZz zNu{9Z0t79(pa3=Y)<^8PrU~L`_7~<|%)EhHMCdK^0=z~Rw0?x2GJe)w5F9gqpkL0Q zyqJ(C4xO^HF=w=vmCfC;H_4VHPa2bda*_PQOis-%wNH7Azh+kyo@jajqtmT4(oSL~ zjnjuqH9(qii1?Mmq_d`|9Lx)It}9cu$QS%lyCyY#CVG0d=+T97F?r{-e8j2~T?ghL zR`RTsPCha&^F~q(`naDq%<4Zje=(}^9ysVG^(x=%VD$ls+n32;`P2bfVEH%wNUWuFB7qg^&9Pam@3|8 zL+m)yphG&C-%SL^Ks@WkI5k5=fu?7a)m;iJrzDicluXO{JoI@_nh#5Pr8LlLcFHa@ zH?_4$V4tU}WYP3P>i>v;R1^+|DV>y+P#G$wXEBwCDcY4J42p~Ppl;M^UOD4Po9m&^ z&8$JmxbUD~OuH&s{i3w|*BgqHt3bgbPj)Os>6TJRY5j4q9js?es=s<#I z2INe8%M3&=2}JCxQ5j(0hyhJeqx;w(zG1S*H8BkeT(1yg*%e%W-^LHuec1e)Yjmhp zIt+MaLiLn<+24`MfYH|r>5*jjwM_uZ>r8a={<1-Y%DmiIwN?6c%sCm~)BpE<%@w0l za6fxA^2J%naQ$K5_-P5I-Pj(K?=PHIf>|-H+ddlbwY2Fu++=ZMR$8!-R+7Mc++Rtp z9aGkiE18NAR+B=1;MyuKl!=R}gC-6Xgc@*Kv5$LtYMcybMu~3xiBt#!wkgibDaOqp z6dP`hg<#0}9?L5b3d1YsH=+XJL(HQr7(j~#q8JCo2iHOi{c>3(&Ha&6m2+WU(3Jt) zY;ey8V$jo2&h|yF{rw2yu0%hx%E&G4pui<}W_8%aWNF3YFYe`{oWH%hKmjPL zE|HFyAJkl2A{`n6VJpO{bnt!Y&^k8yM@h%(HnV~W)fF7O^14p)8GMj65*j^t_o!So8~I z08CL2adAk0#1uEo@nJ?FVldNdk>V@(>Y-(S$AT5mwjbpCnK|dMgT%P@M>in2aLfjE z(Ml)>{1Nifl~7)RS=5)TgbFgt$t5chJT&UnmEd$G(#O;+L!=%2I8{D#8X)R1)~qk& zmXdv^uCSC*VJ--wfv!SoR$1vW1k4L3+t)Y%#bw2cx;O5{rb_86yC?vk(;&m=(clj7La#{I~_3c^(oTj^JnsP@cB)w*RSSWNshH{mX@IG ziZv+_dTyD}A97vQ|UD zxeQjTR;8#7sS28k?~=4)5!3`vfLn#Qj!&9@kc6(jOsXQZ=$wdZ7;C4 zVcNw?e^26S>(4lw?R0+xFeeGo9t8AHNcVd%PfnlCDroTYWzF{MjU%wRJN6#!)D1a0 zX@`TL*`f?Z9eYD{+(t|bW|xCyp-5*jtp^@}M-V+4XNOp9V`>88deCuObF;i1FRO=t zUVS?YwVltK>9c{SgF4F#eQqoL_NZ{g0}J9=WW#~!5oT5i!g-Y5ShTqB)5T+U1do>0 zbFax%aQyC@4}F0(KuCi=keKt#@@ z3x~pgX%(!L7+Kb5gB~w;3)a@Xc0d|`+nTRPV->A~w)I@j+kq&-jC1jGSCn#h1&+OI z7fK`*CAsdTE@fd-f3)B(2fS#0*hs>=wM#YJ()Tc! 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