From 1c1da2778964df3c0923f122ced20b259bd67095 Mon Sep 17 00:00:00 2001 From: Luca Weidlich Date: Thu, 11 Jun 2026 10:24:21 +0200 Subject: [PATCH] =?UTF-8?q?Quellen=20und=20kleine=20=C3=84nderungen=20an?= =?UTF-8?q?=20der=20Prefars?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- bibliography.bib | 74 +++++++++++++++++------- content/Mathematischer_Hintergrund.tex | 76 ++++++++++++++++++------- galton.pdf | Bin 296457 -> 338406 bytes galton.tex | 5 -- images/Galton_Hüpfbild.png | Bin 0 -> 4551 bytes images/Galtonbrett.png | Bin 0 -> 23362 bytes 6 files changed, 111 insertions(+), 44 deletions(-) create mode 100644 images/Galton_Hüpfbild.png create mode 100644 images/Galtonbrett.png diff --git a/bibliography.bib b/bibliography.bib index 58795fb..a9da459 100644 --- a/bibliography.bib +++ b/bibliography.bib @@ -1,22 +1,58 @@ -% Ein Eintrag in der Bibliographie könnte so aussehen: -@book{Beutelspacher2009, - shorthand = {Beu09}, - author = {Beutelspacher, Albrecht}, - editor = {}, - publisher = {Vieweg+Teubner Verlag}, - title = {"Das ist o.B.d.A. trivial!"}, - subtitle = {Tipps und Tricks zur Formulierung mathematischer Gedanken}, - year = {2009}, - edition = {9. Auflage}, - url = {https://doi.org/10.1007/978-3-8348-9599-8}, - note = {Zugriff am 31. Mai 2026} +@unpublished{kosenkova_stochastik_2025, + location = {Universität potsdam}, + title = {Stochastik für das Lehramt}, + url = {https://moodle2.uni-potsdam.de/course/view.php?id=44845}, + shorttitle = {{StoLa}}, + type = {Vorlesung}, + howpublished = {Vorlesung}, + author = {Kosenkova, Dr. Tetiana}, + urldate = {2026-06-11}, + date = {2025}, +} + + +@book{nahrstedt_monte-carlo-methode_2015, + location = {Wiesbaden}, + title = {Die Monte-Carlo-Methode: Beispiele unter Excel {VBA}}, + isbn = {978-3-658-10149-7}, + series = {Essentials}, + shorttitle = {Die Monte-Carlo-Methode}, + publisher = {Springer Fachmedien Wiesbaden}, + author = {Nahrstedt, Harald}, + date = {2015}, + langid = {german}, +} + +@collection{buchter_elementare_2005, + location = {Berlin, Heidelberg}, + title = {Elementare Stochastik: Eine Einführung in die Mathematik der Daten und des Zufalls}, + isbn = {978-3-540-22250-7 978-3-540-27368-4}, + doi = {10.1007/b138982}, + series = {Mathematik für das Lehramt}, + shorttitle = {Elementare Stochastik}, + pagetotal = {452}, + publisher = {Springer-Verlag Berlin Heidelberg}, + editor = {Büchter, Andreas and Henn, Hans-Wolfgang}, + date = {2005}, + langid = {german}, +} + +@online{arndt_brunner_simulation_2025, + title = {Simulation des Galtonbretts}, + url = {https://www.arndt-bruenner.de/mathe/scripts/galtonbrett.htm}, + titleaddon = {Simulation des Galtonbretts}, + author = {{Arndt Brünner}}, + urldate = {2025-06-11}, + date = {2025-11-29}, +} + +@video{statistik_verstehen_beweis_2019, + title = {Beweis des zentralen Grenzwertsatzes von de Moivre-Laplace}, + url = {https://www.youtube.com/watch?v=0_RZUWtlCQM}, + editora = {{Statistik verstehen}}, + editoratype = {collaborator}, + urldate = {2025-06-11}, + date = {2019-10-28}, } -% Beutelspacher2009 ist der Zitationsschlüssel, der in \cite{Beutelspacher2009} verwendet wird. Er sollte eindeutig sein und kann nach Belieben gewählt werden, z.B. durch Kombination von Autorennamen und Jahr. -% Shorthand gibt vor, wie der Eintrag in der Bibliographie angezeigt wird. -% Bei einem Autor sollten die ersten drei Buchstaben des Nachnamens und die letzten zwei Ziffern des Jahres verwendet werden, z.B. Beu09 für Beutelspacher 2009. -% Bei mehreren Autoren sollten die Initialien der Nachnamen der ersten drei Autoren und die letzten zwei Ziffern des Jahres verwendet werden, z.B. ABM20 für einen Eintrag von Autoren A, B und M aus dem Jahr 2020. -% Die anderen Einträge (author, editor, publisher, title, year, edition, url) sollten entsprechend den Informationen des zitierten Werks ausgefüllt werden. -% Es ist hilfreich, die URL anzugeben, wenn das Werk online verfügbar ist, damit Leser leicht darauf zugreifen können. -% Wenn die URL angegeben wird, sollte auch das Zugriffsdatum hinzugefügt werden, um anzugeben, wann die Quelle zuletzt überprüft wurde. diff --git a/content/Mathematischer_Hintergrund.tex b/content/Mathematischer_Hintergrund.tex index 11ca296..9e56ed9 100644 --- a/content/Mathematischer_Hintergrund.tex +++ b/content/Mathematischer_Hintergrund.tex @@ -1,13 +1,29 @@ %"ltex.language": "de-DE" -Dies ist der zentrale Teil des Dokuments und soll (inhaltlich) den größten Teil des Dokuments ausmachen. - \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} +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 \cite{nahrstedt_monte-carlo-methode_2015}. +\begin{definition}[Modell des Galton Brett]{galton}{nach \cite[S. 252-253]{buchter_elementare_2005}} Sei $(\Omega, \mathcal{F}, P)$ ein Wahrscheinlichkeitsraum. \\ Der Fall einer Kugel durch ein Galton-Brett mit $n \in \mathbb{N}$ Reihen wird modelliert durch eine Folge von stochastisch unabhängigen und identisch verteilten Zufallsvariablen $X_1, X_2, \dots, X_n$, wobei $X_i \in \{0, 1\}$. Dabei beschreibt \textbf{$X_i = 1$ den Fall nach rechts} in der $i$-ten Reihe und \textbf{$X_i = 0$ den Fall nach links}. Die Wahrscheinlichkeit sei $P(X_i = 1) = p$ und $P(X_i = 0) = 1-p = q$. Bei einem symmetrischen Brett gilt $p = q = 0.5$. \end{definition} %Bild? +\begin{figure}[htbp] + \centering + % Erstes Bild + \begin{minipage}[t]{0.45\textwidth} + \centering + \includegraphics[width=\linewidth]{./images/Galtonbrett.png} + \caption{Darstellung des Galtonbretts nach \cite{arndt_brunner_simulation_2025}} + \label{fig:galton} + \end{minipage} + \hfill + % Zweites Bild + \begin{minipage}[t]{0.45\textwidth} + \centering + \includegraphics[width=\linewidth]{./images/Galton_Hüpfbild.png} + \caption{Darstellung der Zufallsvariable $X_i$ im Kontext vom Galtonbrett} + \label{fig:galtonXi} + \end{minipage} +\end{figure} Hierbei ist anzumerken, dass jedes $X_i$ Bernoulliverteilt ist, da \[ \Omega = \{1,0\} \quad \text{und} \quad @@ -19,27 +35,44 @@ Hierbei ist anzumerken, dass jedes $X_i$ Bernoulliverteilt ist, da \] Somit lässt sich auch $X_i \sim \mathcal{B}_{0,5}$ schreiben.\\ Um jeden Ausgang des Galton-Brettes durchnummeriert von links nach rechts unterscheiden zu können, definieren wir uns eine weitere Zufallsvariable Sn wie folgt: -\begin{definition}[Zufallsvariable $S_n$]{Sn} +\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. +% Def +Betrachten wir nun den Binomialkoeffizienten in Bezug auf $S_n$, beschreibt dieser exakt die Anzahl der möglichen Pfade durch das Galton-Brett, bei denen die Kugel von $n$ Reihen genau $k$-mal nach rechts (und somit $(n-k)$-mal nach links) fällt. Da jeder dieser einzelnen Pfade aufgrund der Unabhängigkeit der Entscheidungen die Wahrscheinlichkeit $p^k (1-p)^{n-k}$ besitzt, ergibt sich die Gesamtwahrscheinlichkeit für das Fach $k$ durch Multiplikation. Daher rührt die Binomialverteilung: +\begin{definition}[Binomialmodell]{binModell}{\cite{kosenkova_stochastik_2025}} + Wiederholt man ein Bernoulli-Experiment mit Erfolgswahrscheinlichkeit $p \in [0,1]$ $n$-mal, und interessiert sich nur für die Anzahl der erfolgreichen Experimente, so wählt man + \[ + \Sigma = \{0,1,...,n\}. + \] + In diesem Modell ist + \[ + Bin_{n,p}(\{k\}) := \binom{n}{k} p^k (1-p)^{n-k}, \quad k=0,1,..,n + \] + eine Zähldichte. +\end{definition} +\begin{definition}{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} Anschließend ist folgendes zu bemerken: \begin{satz}[Verteilung der Endposition]{satz:binomialverteilung} - Die Zufallsvariable $S_n$, ist binomialverteilt mit den Parametern $n$ und $p$. + Die Wahrscheinlichkeitsverteilung der Zufallsvariable $S_n$, lässt sich durch die Binomialverteilung beschreiben.\\ Daher gilt $S_n \sim \mathcal{B}in_{n,p}$ \end{satz} \begin{proof} Nach \cref{def:galton} 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. - %Ref Cosi - Daraus folgt, dass das Galton-Brett mit der Zufallsvariable $S_n$ eine Binomialmodell darstellt und somit die Binomialverteilung nach Definition eine Zähldichte definiert. Daher gilt: $S_n \sim \mathcal{B}in_{n,p}$ + Daraus folgt, dass das Galton-Brett mit der Zufallsvariable $S_n$ eine Binomialmodell \cref{def:binModell} darstellt und somit die Binomialverteilung nach Definition eine Zähldichte definiert \cref{def:binZähldichte}. Daher gilt: $S_n \sim \mathcal{B}in_{n,p}$ \end{proof} -Anhand des Galton-Brettes lässt sich nun wie bereits beschrieben leicht erkennen, dass bei $n\rightarrow\infty$ Kugeln sich die Verteilung der Kugeln in der Auffangung der Verteilung der Glockenkurve der Normalverteilung annähert. %"Verteilung" Fachlick Korrekt? -\begin{satz}[Zentraler Grenzwertsatz von de Moivre-Laplace]{satz:moivrelaplace} +Für große $n$ wird die direkte Berechnung der Binomialverteilung sehr aufwendig. An dieser Stelle greift der Zentrale Grenzwertsatz, der besagt, dass sich die Binomialverteilung für große $n$ der Normalverteilung annähert. 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} 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 $S_n$ genau den Wert $k$ annimmt, durch die Dichtefunktion der Normalverteilung annähern: \[ P(S_n = k) = \binom{n}{k} p^k q^{n-k} \approx \frac{1}{\sqrt{2\pi npq}} e^{\left(-\frac{(k-np)^2}{2npq}\right)} \] \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.\\ 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 3 Approximationen werden in diesem Beweis als wahr angenommen, jedoch in der Folgenden Arbeit weiter analysiert und bewiesen. \textbf{1. Anwendung der Stirling-Formel:} \\ @@ -63,7 +96,7 @@ Anhand des Galton-Brettes lässt sich nun wie bereits beschrieben leicht erkenne \textbf{2. Substitution und Approximation der Wurzeln:} \\ Um das breiter werden der Verteilung und das Abwandern des Erwartungswertes zu verhindern standardisieren wir die Zufallsvariable, indem wir \[ - Z_n=\frac{S_n-E(X_n)}{\sigma(S_n)} = \frac{k-np}{\sqrt{npq}} + Z_n=\frac{S_n-E(S_n)}{\sigma(S_n)} = \frac{k-np}{\sqrt{npq}} \] definieren. Nun nimmt $Z_n$ Werte $z=\frac{k-np}{\sqrt{npq}}$ an, nach $k$ und nach $n-k$ umgestellt heißt das \begin{align*} @@ -112,13 +145,13 @@ Anhand des Galton-Brettes lässt sich nun wie bereits beschrieben leicht erkenne \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 Taylor-Approximation besagt: $ln(1-\alpha) \approx \alpha-\frac{\alpha^2}{2}+O(\alpha^3)$ Hierbei steht $O(\alpha^3)$ für die Landau-Notation. + \cref{def:taylorpolynom} beweist die Taylor-Approximation für den Logarithmus: $ln(1-\alpha) \approx \alpha-\frac{\alpha^2}{2}+O(\alpha^3)$ 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 $|\alpha| < 1$. Das heißt, dass $\forall z \in \mathbb{R}: |z\cdot\beta| \overset{n\rightarrow\infty}{<}1$. (Für $z\cdot\gamma$ analog)\\ + Um diese Approximation zu verwenden, muss sicher sein, dass $|\alpha| < 1$. Das heißt, dass $\forall z \in \mathbb{R}: |z\cdot\beta| \overset{n\rightarrow\infty}{<}1$. (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 < 1 @@ -133,13 +166,13 @@ Anhand des Galton-Brettes lässt sich nun wie bereits beschrieben leicht erkenne und für den anderen ln analog: \begin{align*} \ln\left(1 - z\sqrt{\frac{p}{nq}}\right) &\approx -z\sqrt{\frac{p}{nq}} - \frac{\left(z\sqrt{\frac{p}{nq}}\right)^2}{2} + O((-z\gamma)^3) \\ - &= -z\sqrt{\frac{p}{nq}} - \frac{z^2 p}{2nq} + O((z\gamma)^3) &&\text{- fällt wegen O weg} + &= -z\sqrt{\frac{p}{nq}} - \frac{z^2 p}{2nq} + O((z\gamma)^3) &&\text{"-" fällt wegen O weg} \end{align*} Setzen wir dies nun zurück in \cref{eq5:3} ein, erhalten wir: \begin{align} - &(-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) \nonumber \\ - &=(-np - z\sqrt{npq})(z\sqrt{\frac{q}{np}} - \frac{z^2 q}{2np} + O((z\beta)^3)) \nonumber \\ &\quad + (-nq + z\sqrt{npq}))(-z\sqrt{\frac{p}{nq}} - \frac{z^2 p}{2nq} + O((z\gamma)^3)) \nonumber \\ - &=(-npz\sqrt{\frac{q}{np}}+\frac{npz^2q}{2np}-z^2\sqrt{npq}\sqrt{\frac{q}{np}} + O((z\beta)^3)) \nonumber\\ &\quad + (nqz\sqrt{\frac{p}{nq}}+\frac{nqz^2p}{2nq}-z^2\sqrt{npq}\sqrt{\frac{p}{nq}} + O((z\gamma)^3))\nonumber \\ &\quad + (-np - z\sqrt{npq}) \cdot O((z\beta)^3) + (-nq + z\sqrt{npq}) \cdot O((z\gamma)^3)\label{eq6:1}\\ + &(-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)+O(z^3) \nonumber \\ + &=(-np - z\sqrt{npq})(z\sqrt{\frac{q}{np}} - \frac{z^2 q}{2np} + O((z\beta)^3)) \nonumber \\ &\quad + (-nq + z\sqrt{npq}))(-z\sqrt{\frac{p}{nq}} - \frac{z^2 p}{2nq} + O((z\gamma)^3)) + O(z^3) \nonumber \\ + &=(-npz\sqrt{\frac{q}{np}}+\frac{npz^2q}{2np}-z^2\sqrt{npq}\sqrt{\frac{q}{np}} + O((z\beta)^3)) + O(z^3) \nonumber\\ &\quad + (nqz\sqrt{\frac{p}{nq}}+\frac{nqz^2p}{2nq}-z^2\sqrt{npq}\sqrt{\frac{p}{nq}} + O((z\gamma)^3))\nonumber \\ &\quad + (-np - z\sqrt{npq}) \cdot O((z\beta)^3) + (-nq + z\sqrt{npq}) \cdot O((z\gamma)^3)\label{eq6:1}\\ &=(-z\sqrt{\frac{(np)^2q}{np}}+\frac{z^2q}{2}-z^2\sqrt{\frac{npq^2}{np}}+O((z\beta)^3)) \nonumber \\ &\quad + (z\sqrt{\frac{(nq)^2p}{nq}}+\frac{z^2p}{2}-z^2\sqrt{\frac{np^2q}{nq}} + O((z\gamma)^3)) \nonumber \\ &\quad + (-np - z\sqrt{npq}) \cdot O((z\beta)^3) + (-nq + z\sqrt{npq}) \cdot O((z\gamma)^3) \label{eq6:2}\\ &=(-z\sqrt{npq}-z^2q+\frac{z^2q}{2} + O((z\beta)^3)) + (z\sqrt{npq}-z^2p+\frac{z^2p}{2} + O((z\gamma)^3))\nonumber \\ &\quad + (-np - z\sqrt{npq}) \cdot O((z\beta)^3) + (-nq + z\sqrt{npq}) \cdot O((z\gamma)^3)\label{eq6:3} \\ &\approx (-z\sqrt{npq}-\frac{z^2q}{2}) + (z\sqrt{npq}-\frac{z^2p}{2})\label{eq6:4}\\ @@ -151,7 +184,7 @@ Anhand des Galton-Brettes lässt sich nun wie bereits beschrieben leicht erkenne \item[(\ref{eq6:1})] Distributivgesetz \item[(\ref{eq6:2})] $np$ und $nq$ mithilfe des Quadrates in die Wurzel ziehen, sowie Kürzen und Wurzeln zusammenfassen. Alle Terme mit $z^3$ werden von $O(z^3)$ nach dessen Definition "absorbiert". \item[(\ref{eq6:3})] $np$ und $nq$ in den Wurzeln Kürzen und Kommutativgesetz anwenden - \item[(\ref{eq6:4})] Die Landau Terme ($O(z^3)$) können vernachlässigt werden, da diese Terme für $n\rightarrow\infty$ gegen 0 gehen. Betrachte Hierzu \cref{lem:restterme} und \cref{prop:restterme2}. + \item[(\ref{eq6:4})] Die Landau Terme können vernachlässigt werden, da diese Terme für $n\rightarrow\infty$ gegen 0 gehen. Betrachte Hierzu \cref{lem:restterme} und \cref{prop:restterme2}. \item[(\ref{eq6:5})] $-z\sqrt{npq}+z\sqrt{npq}=0$ und Distributivgesetz (invers) \item[(\ref{eq6:6})] $p+q=1$ da $q = 1-p$ definiert wurde. \end{itemize} @@ -166,7 +199,7 @@ Anhand des Galton-Brettes lässt sich nun wie bereits beschrieben leicht erkenne \item[(\ref{eq7:1})] Rücksubstitution der Standardisierung. \item[(\ref{eq7:2})] Der Bruch wurde quadriert und zusammengefasst. \end{itemize} - Daher konvergiert eine Binomialverteilte Zufallsvariable (für $n\rightarrow\infty$) gegen die Standardnormalverteilung. + 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 $z \in \mathbb{R}$ fest gewählt. Für die Restterme der Taylor-Entwicklung des Logarithmus multipliziert mit ihren jeweiligen Vorfaktoren gilt für $n \rightarrow \infty$: @@ -194,12 +227,15 @@ Anhand des Galton-Brettes lässt sich nun wie bereits beschrieben leicht erkenne Für den zweiten Term mit $\gamma$ erfolgt der Beweis völlig analog, womit die Summe beider Terme ebenfalls gegen $0$ konvergiert. \end{proof} -Folgende Proposition ich leicht aus dem oberen Beweis ableitbar (aufgrund der Abschätzung mit $n^{-3/2}$) und wird daher nicht separat bewiesen. +Folgende Proposition ich leicht aus dem oberen Beweis ableitbar (aufgrund der Abschätzung mit $n^{-3/2}$ und der Betrachtung von z als beliebig aber fest) und wird daher nicht separat bewiesen. \begin{proposition}[Asymptotisches Verschwinden der Taylor-Restterme ohne Präfix]{restterme2} Seien $p \in (0,1)$, $q = 1-p$ und $z \in \mathbb{R}$ fest gewählt. Für die Restterme der Taylor-Entwicklung des Logarithmus gilt für $n \rightarrow \infty$: \[ O((z\beta)^3) + O((z\gamma)^3) \xrightarrow{n \rightarrow \infty} 0 \] + \[ + O(z^3) \xrightarrow{n \rightarrow \infty} 0 + \] wobei $\beta = \sqrt{\frac{q}{np}}$ und $\gamma = \sqrt{\frac{p}{nq}}$ definiert sind. \end{proposition} diff --git a/galton.pdf b/galton.pdf index 4e3104f1f6cf91ba040085d0526b5a1aeb6eafb4..4695fa461389935e2316dc8491b2056fc7dc089a 100644 GIT binary patch delta 150428 zcmcG#by!tzw>7*;5h)cVB}7^}1Vkwb>69*Mq#Fblf}(&3lG5E0lG3Pvga{HMAsx~! z4d2|q=lkP1*BjqC@An=rxJ38f_gd?|=Nxm4F()CfuPh9Z20X{8#H7wec!h=U-^X`# zbFnbB$M+gt(|Z_UAVAWz!#+W4Z5Hoo!+)Xm=Er9)Qq;KvHp-U+&yf}j?GcOKqy+yP zmoKx1X1X=}Z1VLVn@k#UJ2b1t(S&#VQnUL)VA_8jiH z5F0(FOkZf--7&v^p?EszC$iA^6N5(O=P%yJX``t@0 zuim5kqE-|@pkMU*QfnE<@r~D6242s_WxwQ*{NQ9$Jbr%uMc8%P`1_SkB;F(RQo|)H zsnN-6Js+_=Gpr+*j*bo5OSc`!R#-dwF1k%KIWU`Pd4%ZT1-}uwdYSCj^5q$*c!(rfiqzI!TC}llHF}umj+@$%ns|f$$$d66`)O`A 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