From 75b03ac7d66b089f6239fd385b8766b7244a77d2 Mon Sep 17 00:00:00 2001 From: Luca Weidlich Date: Tue, 21 Jul 2026 09:09:03 +0200 Subject: [PATCH] letzte Typos korrigiert --- content/Einleitung.tex | 2 +- content/Mathematischer_Hintergrund.tex | 12 ++++++------ galton.pdf | Bin 979010 -> 978926 bytes material/ab1.pdf | Bin 110039 -> 110039 bytes material/ab1.tex | 6 +++--- material/ab3.pdf | Bin 293717 -> 292735 bytes material/ab3.tex | 16 ++++++++-------- 7 files changed, 18 insertions(+), 18 deletions(-) diff --git a/content/Einleitung.tex b/content/Einleitung.tex index 3b13206..15295a8 100644 --- a/content/Einleitung.tex +++ b/content/Einleitung.tex @@ -2,7 +2,7 @@ Das Galtonbrett zählt zu den bekanntesten Demonstrationsmodellen der Wahrschein Aus mathematischer Sicht bildet das Galtonbrett die Binomialverteilung ab, wobei sich die Verteilung mit zunehmender Anzahl an Stiftreihen einer Normalverteilung annähert. Das Verhalten steht in engem Zusammenhang mit dem zentralen Grenzwertsatz. Zur Beschreibung dieser Annäherung spielen außerdem die Fakultätsfunktion sowie die zugehörige Approximation durch die Stirling-Formel eine wichtige Rolle. -Das Galtonbrett stellt einen hohen didaktischen Wert dar, indem es ein anschauliches Experiment zu Grundkonzepten der Stochastik und Analysis und deren Verknüpfung liefert. Das Modell eignet sich besonders für den Einsatz in der Sekundarstufe II in Mathe-AGs und -clubs, in denen über den Unterricht hinausgehende Inhalte behandelt werden können. Insbesondere die Verbindung von praktischem Experiment, Simulation und theoretischer Herleitung ermöglicht einen vertiefern Zugang zur Wahrscheinlichkeitstheorie. +Das Galtonbrett stellt einen hohen didaktischen Wert dar, indem es ein anschauliches Experiment zu Grundkonzepten der Stochastik und Analysis und deren Verknüpfung liefert. Das Modell eignet sich besonders für den Einsatz in der Sekundarstufe II in Mathe-AGs und -clubs, in denen über den Unterricht hinausgehende Inhalte behandelt werden können. Insbesondere die Verbindung von praktischem Experiment, Simulation und theoretischer Herleitung ermöglicht einen vertieften Zugang zur Wahrscheinlichkeitstheorie. Ziel der Ausarbeitung ist es, die mathematischen Grundlagen des Galtonbretts auf Hochschulniveau darzustellen und deren Zusammenhänge verständlich herzuleiten. Anschließend werden diese Inhalte auf das mathematische Vorwissen der Klassenstufe 10 bis 12 übertragen und didaktisch aufbereitet. Die entstehenden Arbeitsmaterialien ermöglichen den Schüler*innen einen vertieften Einblick in die mathematischen Hintergründe des Galtonbretts. diff --git a/content/Mathematischer_Hintergrund.tex b/content/Mathematischer_Hintergrund.tex index f1622fa..b602405 100644 --- a/content/Mathematischer_Hintergrund.tex +++ b/content/Mathematischer_Hintergrund.tex @@ -90,7 +90,7 @@ Es ist mathematisch gesehen sehr schön zu sehen, wie eine relativ einfache disk \lim_{n \to \infty} \mathop{\max}\limits_{k:|z|\leq c} \left|\frac{P(S_n=k)}{f_\mathcal{N}(\mu, \sigma^{2})}(k)\right| = 1 \] \end{satz} -In der Literatur wird allgemein vom Satz von Moivre-Laplace gesprochen, wenn die Approximation der Binomial- durch die Normalverteilung gemeint ist \cite{statistik_verstehen_beweis_2019}. Mathematisch präzise muss jedoch zwischen dem globalen und dem lokalen Grenzwertsatz unterschieden werden \cite{georgii_stochastik_2015, kosenkova_stochastik_2025}. Wir beweisen im Folgenden den lokalen Grenzwertsatz, der das (lokale) Annähern des ``oberen Randes'' des Histogramms der Binomialverteilung an die Glockenkurve beschreibt \cite{georgii_stochastik_2015}. Hierbei wird \emph{nicht} die Approximation der Binomial durch die Normalverteilung bewiesen. +In der Literatur wird allgemein vom Satz von Moivre-Laplace gesprochen, wenn die Approximation der Binomial- durch die Normalverteilung gemeint ist \cite{statistik_verstehen_beweis_2019}. Mathematisch präzise muss jedoch zwischen dem globalen und dem lokalen Grenzwertsatz unterschieden werden \cite{georgii_stochastik_2015, kosenkova_stochastik_2025}. Wir beweisen im Folgenden den lokalen Grenzwertsatz, der das (lokale) Annähern des ``oberen Randes'' des Histogramms der Binomialverteilung an die Glockenkurve beschreibt \cite{georgii_stochastik_2015}. Hierbei wird \emph{nicht} die Approximation der Binomial- durch die Normalverteilung bewiesen. \begin{proof} Der Beweis folgt zum großen Teil dem Beweis von \cite{statistik_verstehen_beweis_2019} und wurde von den Autoren aufgrund der Konkretisierung des Satzes \cite{georgii_stochastik_2015} weiter konkretisiert.\\ Der Beweis basiert im Wesentlichen auf drei Approximationen: Der Stirling-Formel für die Fakultäten, der Vereinfachung der Wurzelausdrücke für große $n$, sowie der Taylor-Entwicklung des natürlichen Logarithmus zur Herleitung der Exponentialfunktion. Diese drei Approximationen werden in diesem Beweis als wahr angenommen, jedoch in der folgenden Arbeit weiter analysiert und bewiesen.\\ @@ -289,7 +289,7 @@ Die Gamma-Funktion berechnet wie folgt die Fakultät in $\mathbb{N}$: Ein Beweis hierzu befindet sich in \cite{forster_analysis_2023}. \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. +Die Stirling-Approximation ist eine mathematische Näherung zur Berechnung der Fakultät einer Zahl. Die Approximation ist vor allem 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 \[ @@ -366,7 +366,7 @@ Die Stirling-Approximation ist eine mathematische Näherung zur Berechnung der F &\left(d_n-\frac{1}{12n}\right)-\left(d_{n+1}-\frac{1}{12(n+1)}\right)\\ <\quad0\quad<\quad&\left(d_n-\frac{1}{12n+1}\right)-\left(d_{n+1}-\frac{1}{12(n+1)+1}\right) \end{align*} - folgt, dass die Folge $\left\{d_n-\frac{1}{12n+1}\right\}$ monton fallend, die Folge $\left\{d_n-\frac{1}{12n}\right\}$ monoton wachsend ist. Beide Folgen haben dabei den Grenzwert $c:=\lim_{n\rightarrow\infty} d_n$. + folgt, dass die Folge $\left\{d_n-\frac{1}{12n+1}\right\}$ monoton fallend, die Folge $\left\{d_n-\frac{1}{12n}\right\}$ monoton wachsend ist. Beide Folgen haben dabei den Grenzwert $c:=\lim_{n\rightarrow\infty} d_n$. \begin{align*} d_n-\frac{1}{12n}\quad&<\quad c\quad<\quad d_n-\frac{1}{12n+1}\\ c+\frac{1}{12n+1}\quad&<\quad d_n\quad<\quad c+\frac{1}{12n} @@ -488,7 +488,7 @@ Der Beweis entspricht im wesentlichen dem 2006 von Wolfgang Kössler geführten %\end{proof} \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. Da $\ln(1-x)$ sogar unendlich oft differenzierbar ist, ist diese Aussage trivial. +Um eine Taylorapproximation 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. Da $\ln(1-x)$ sogar unendlich oft differenzierbar ist, ist diese Aussage trivial. \begin{proposition}{dreimalableiten} $f(x)=\ln(1-x)$ ist mindestens dreimal differenzierbar. \end{proposition} @@ -509,7 +509,7 @@ Bevor wir die Approximation nachweisen, ist eine formale Definition des Satzes \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+1$-ten Ableitung von $f$ an der Stelle $\xi$. - 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". + 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. "Algorithmus 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} @@ -560,7 +560,7 @@ 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, 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. + 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 Lagrange-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] diff --git a/galton.pdf b/galton.pdf index c63990c758122a76ba493dfee67f6eb9607784a4..b190d30b7cb8041f1baf913c9d22e0ff627f58ae 100644 GIT binary patch delta 50594 zcmV)2K+M0w;5+W$JAi}%v;uoM0XLVix(gMTkvRgSfB1V^oq(5mnt7XfN*UfpJ)W2s zo3F68aa2!X9CE&9?tO&4LJ$2X?hT!O(?sVy*JKShfm62U_>?qrE(}R3p*e*lvxC>zfvC_@o{T?2KJq!(3Ya45&+mqu;`Lh5Ck)R1+bO!y_&8=FH-&?TlBhDNYL`gVtsi1+kxB z?c#@4f!|k+egIYlnj+ddPq-3oL@|K79ODEff0_{j&o18IximXG;aWucWwmiP+Et`p znL}IUD|Gdku@sQa|C!r)YlR!bu~l$yD-d`?s^N}BE4bqjwf2;#d^#v_XS{<`aKK2G z1*he*m4Abs?jd7>1pMSsopl>BIXsiUk9+^ide?DGGs$OisNBKWT3I9o<;+Ov(F`)pt{AdS3e6H9OUnKcD zC;*)mZuL>&7`0QHa2c?*=t+kq26qh8@o^O45`@J@h$D%3OC%9nnyGY{p|h>|vJ17{ 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Warmwerden berechnen wir: @@ -364,7 +364,7 @@ \noindent \begin{minipage}[t]{0.62\textwidth}\vspace{0pt} \begin{definitionbox}[Galtonbrett] - Das \textbf{Galtonbrett} ist ein mechanisches Demonstrationsgerät aus der Stochstik, bei welchem kleine Kugeln auf horizontal und dreieckig angeordnete Stäbchen fallen gelassen wird (hier zweidimenional von vorne dargestellt). Bei jedem Aufprall auf ein Stäbchen teilt sich ihr Weg: entweder nach links, oder nach rechts. + Das \textbf{Galtonbrett} ist ein mechanisches Demonstrationsgerät aus der Stochastik, bei welchem kleine Kugeln auf horizontal und dreieckig angeordnete Stäbchen fallen gelassen wird (hier zweidimensional von vorne dargestellt). Bei jedem Aufprall auf ein Stäbchen teilt sich ihr Weg: entweder nach links, oder nach rechts. \end{definitionbox} \end{minipage} \hfill @@ -476,7 +476,7 @@ \item Dir ist sicherlich schon aufgefallen, dass die meisten Pfade in die Mitte führen. Untersuche nun die Anzahl der Pfade zu jedem der Stäbchen und vergleiche diese mit dem Pascalschen Dreieck. Was fällt dir auf? \vspace{85pt} - \item Gib die Verteilung des Galtonbretts mithilfe einer Zufallsvariabel $X$ an. Wie könnte diese mit dem pascalschen Dreieck zusammenhängen? + \item Gib die Verteilung des Galtonbretts mithilfe einer Zufallsvariable $X$ an. Wie könnte diese mit dem pascalschen Dreieck zusammenhängen? \end{enumerate} \label{LastPage} diff --git a/material/ab3.pdf b/material/ab3.pdf index 1252bccbb83b43ab1d830c1d211b53073feffa2c..a7222faa35da04009345a98501c239fdb797b2bb 100644 GIT binary patch delta 26785 zcmV)9K*hh+_7VT?5s--sOk6N9Fh^W4Ffdb*;$;FgGnZj-0Th3=8_ALtx$)j#QAaqm zkW%|h(_tP>Pk0Uv6T`s~GjniYgB=D8l0y}5dJuk02@bl}YP=G~RT4!NlTv0<=1-Xi>ME$Jcqt-R4zTbukFivIGaB4S!xfhUir#| zw)*KZ_qpXB#(8tEyURPsd+R4Q*bXlfl96vFyjm56x0<--%ZBv-VM|PKmH5UY<4`m8 z2p^1wjX!?_nC&+l`|R`kcYk>MS0xbgKQvTsMp$i4K$c6NDn#L>vB-A$JAwci%?7v$ zX&p8g0XiaUR>UDp@tlG{|sBo7ua4eP<<%5u+ZB`2_ z6;f}fc{o^Y1eCED6&?lAqrt#dSL;a54eWTOC7m#X*A3;z)Vg!j>6Gd4nH!ZA!5gFx z2oFhzse-}8F6{m7$IOqD#;OyM77|hE?b=HvOi(meR%dagUh42Vn&}sk9e>7KEU9qV9HPB>Lv_ld81Xx>>3JV7+Osk~|OOPvL!GwH19pd8k z#&DIfe$CMS`r))v0!jW&NM*GY#zH5RJtCD0QUupX`G}iOU)YBuo zsvMF?&B)ZnwlSYZ-6rNF#v~i%Gw z${tP?A6bw^BoV>?Cl60sYy?0=SV8(3<%BxGlh#Ev39s$;+uyf$%%w|$q?e@ER-s9Q z0k2O!6>iaC3F1(&%>+`##`qVGdEn~69ZVmO-9ZRIT=cpwG 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