From 8ef15ae6b2e47d268852e585ee98d078b0d88ff6 Mon Sep 17 00:00:00 2001 From: Luca Weidlich Date: Tue, 30 Jun 2026 16:37:29 +0200 Subject: [PATCH] Rechtschreibungszeug --- content/Mathematischer_Hintergrund.tex | 79 ++++++++++++++----------- galton.pdf | Bin 480692 -> 446741 bytes 2 files changed, 44 insertions(+), 35 deletions(-) diff --git a/content/Mathematischer_Hintergrund.tex b/content/Mathematischer_Hintergrund.tex index fedfe8a..b13ce2f 100644 --- a/content/Mathematischer_Hintergrund.tex +++ b/content/Mathematischer_Hintergrund.tex @@ -1,6 +1,6 @@ %"ltex.language": "de-DE" \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 \cite{nahrstedt_monte-carlo-methode_2015}. +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 mithilfe einer 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 $X_i \in \{0, 1\}$. @@ -34,11 +34,11 @@ Hierbei ist anzumerken, dass jedes $X_i$ Bernoulliverteilt ist, da \end{cases} \] 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: +Um jeden Ausgang des Galton-Brettes durchnummeriert von links nach rechts unterscheiden zu können, definieren wir uns eine weitere Zufallsvariable $S_n$ wie folgt: \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} \[ @@ -58,6 +58,7 @@ Betrachten wir nun den Binomialkoeffizienten in Bezug auf $S_n$, beschreibt dies \] eine Zähldichte. \end{definition} +Im Fall des Galtonbrettes können wir $\Sigma = S_n$ wählen. \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} @@ -68,9 +69,14 @@ Anschließend ist folgendes zu bemerken: \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. - 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: $S_n \sim \mathcal{B}in_{n,p}$ + 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} -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}. +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. +%Normalverteilung als Stetig Dichte einfphren +\begin{definition}[Normalverteilung] + Inhalt... +\end{definition} +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: \[ @@ -79,7 +85,7 @@ Für große $n$ wird die direkte Berechnung der Binomialverteilung sehr aufwendi \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. + 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. \textbf{1. Anwendung der Stirling-Formel:} \\ Wir beginnen mit der Definition der Binomialwahrscheinlichkeit: @@ -96,32 +102,35 @@ Für große $n$ wird die direkte Berechnung der Binomialverteilung sehr aufwendi Im Folgenden werden die nummerierten Schritte der Gleichungsketten erläutert: \begin{itemize} \item[(\ref{eq1:e})] Hier heben sich die $e$-Potenzen heraus, da $\left(\frac{n}{e}\right)^n = n^n \cdot e^{-n}$ sowie $e^n / (e^k e^{n-k}) = 1$ gilt. - \item[(\ref{eq1:aufteilen})] Hier lässt sich die Gleichung intelligent aufteilen und etwas umstellen durch die Potenzgesetze, da gilt: $n^n = n^k \cdot n^{n-k}$ Außerdem lässt sich einmal $\sqrt{2\pi}$ Kürzen. - \item[(\ref{eq1:zusammenfassen})] Ist das Resultat nach dem Sortieren der Brüche anhand der Exponenten + \item[(\ref{eq1:aufteilen})] Hier lässt sich die Gleichung intelligent aufteilen und etwas umstellen durch die Potenzgesetze, da gilt: $n^n = n^k \cdot n^{n-k}$. Außerdem lässt sich einmal $\sqrt{2\pi}$ Kürzen. + \item[(\ref{eq1:zusammenfassen})] Ist das Resultat nach dem Sortieren der Brüche anhand der Exponenten. \end{itemize} \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 + Um das Breiter werden der Verteilung und das Abwandern des Erwartungswertes zu verhindern standardisieren wir die Zufallsvariable, indem wir \[ - Z_n=\frac{S_n-E(S_n)}{\sigma(S_n)} = \frac{k-np}{\sqrt{npq}} + Z_n=\frac{S_n-E(S_n)}{\sigma(S_n)} \] - definieren. Nun nimmt $Z_n$ Werte $z=\frac{k-np}{\sqrt{npq}}$ an, nach $k$ und nach $n-k$ umgestellt heißt das + definieren. Nun nimmt $Z_n$ Werte $z=\frac{k-np}{\sqrt{npq}}$ an, nach $k$ umgestellt heißt das + \begin{align*} + k = np + z\sqrt{npq} + \end{align*} + Multipliziert man diese Gleichung mit $-(1)$ und addiert man $n$ erhält man einen ähnlichen Ausdruck für $n-k$: \begin{align*} - k = np + z\sqrt{npq} \\ n-k = nq - z\sqrt{npq} \end{align*} Nun betrachten wir $k \cdot (n-k)$ für $n \rightarrow \infty$: \begin{align} k(n-k) &= (np + z\sqrt{npq})\cdot(nq - z\sqrt{npq}) \label{eq3:einsetzen}\\ &= n^2pq + (nz\sqrt{nqp}(q-p)-z^2npq) \label{eq3:umstellen}\\ - &\overset{n\rightarrow\infty}{=}n^2pq + O(n) \label{eq3:absch} + &\overset{n\rightarrow\infty}{=}n^2pq\label{eq3:absch} \end{align} Im Folgenden werden die nummerierten Schritte der Gleichungsketten erläutert: \begin{itemize} - \item[(\ref{eq3:einsetzen})] Formeln für $k$ und $n-k$ einsetzen - \item[(\ref{eq3:umstellen})] Distributivgesetz anwenden und nach Potenz von $n$ sortieren + \item[(\ref{eq3:einsetzen})] Die Formeln für $k$ und $n-k$ einsetzen. + \item[(\ref{eq3:umstellen})] Das Distributivgesetz anwenden und nach der Potenz von $n$ sortieren. \item[(\ref{eq3:absch})] $n^2$ geht für $n\rightarrow\infty$ schneller nach unendlich als die linearen Restterme. Folglich sind diese für die Abschätzung gegen unendlich vernachlässigbar. \end{itemize} - Setzen wir dieses Ergebnis nun in das Zwischenergebnis von \cref{eq1:zusammenfassen} ein erhalten wir folgendes: + Setzen wir dieses Ergebnis nun in das Zwischenergebnis von \cref{eq1:zusammenfassen} ein, so erhalten wir folgendes: \begin{align} P(S_n = k) &= \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} \nonumber \\ &\approx \frac{1}{\sqrt{2\pi}}\sqrt{\frac{n}{n^2pq}} \left(\frac{np}{k}\right)^k \left(\frac{nq}{n-k}\right)^{n-k} \label{eq4:1}\\ @@ -130,13 +139,13 @@ Für große $n$ wird die direkte Berechnung der Binomialverteilung sehr aufwendi \end{align} Im Folgenden werden die nummerierten Schritte der Gleichungsketten erläutert: \begin{itemize} - \item[(\ref{eq4:1})] Approximation verwenden - \item[(\ref{eq4:2})] $n$ Kürzen und Potenz-/Wurzelgesetze anwenden um Bruch zu vereinfachen - \item[(\ref{eq4:3})] Kehrwert der Brüche Bilden und Wurzeln zusammenfassen + \item[(\ref{eq4:1})] Die Approximation verwenden. + \item[(\ref{eq4:2})] $n$ Kürzen und die Potenz-/Wurzelgesetze anwenden, um den Bruch zu vereinfachen. + \item[(\ref{eq4:3})] Den Kehrwert der Brüche bilden und die Wurzeln zusammenfassen. \end{itemize} - Es ist bereits der korrekte konstante Faktor der Gaußschen Glockenkurve erkennbar.\\ + Es ist bereits der korrekte konstante Faktor der gaußschen Glockenkurve erkennbar.\\ \textbf{3. Taylor-Approximation des exponentiellen Teils:} \\ - Den restlichen Term formen wir um, indem wir ihn logarithmieren um im Anschluss die Approximationsformel für den Logarithmus verwenden zu können. Diese wird ebenso wie die Stirling-Formel im Anschluss bewiesen. + Den restlichen Term formen wir um, indem wir ihn logarithmieren, um im Anschluss die Approximationsformel für den Logarithmus verwenden zu können. Diese wird ebenso wie die Stirling-Formel im Anschluss bewiesen. Hierzu betrachten wir folgendes zuerst einzeln: \begin{align} &\ln\left( \left(\frac{k}{np}\right)^{-k} \left(\frac{n-k}{nq}\right)^{-n+k} \right) \label{eq5:1} \\ @@ -146,13 +155,13 @@ Für große $n$ wird die direkte Berechnung der Binomialverteilung sehr aufwendi \end{align} Im Folgenden werden die nummerierten Schritte der Gleichungsketten erläutert: \begin{itemize} - \item[(\ref{eq5:1})] logarithmieren + \item[(\ref{eq5:1})] Den Logarithmus anwenden. \item[(\ref{eq5:2})] Logarithmengesetze für Potenzen anwenden. \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} - \cref{def:taylorpolynom} beweist die Taylor-Approximation für den Logarithmus: $ln(1-\alpha) \approx -\alpha-\frac{\alpha^2}{2}+O(\alpha^3)$ und $ln(1+\alpha) \approx \alpha-\frac{\alpha^2}{2}+O(\alpha^3)$. Hierbei steht $O(\alpha^3)$ für die Landau-Notation. - Wir definieren + Die \cref{def:taylorpolynom} beweist die Taylor-Approximation für den Logarithmus: $ln(1-\alpha) \approx -\alpha-\frac{\alpha^2}{2}+O(\alpha^3)$ und $ln(1+\alpha) \approx \alpha-\frac{\alpha^2}{2}+O(\alpha^3)$. Hierbei steht $O(\alpha^3)$ für die Landau-Notation. + Wir definieren: \[ \beta := \sqrt{\frac{q}{np}}; \quad \gamma := \sqrt{\frac{p}{nq}} @@ -188,14 +197,14 @@ Für große $n$ wird die direkte Berechnung der Binomialverteilung sehr aufwendi \end{align} %(-np - z\sqrt{npq}) \cdot O((z\beta)^3) + (-nq + z\sqrt{npq}) \cdot O((z\gamma)^3) Im Folgenden werden die nummerierten Schritte der Gleichungsketten erläutert: \begin{itemize} - \item[(\ref{eq6:1})] Distributivgesetz + \item[(\ref{eq6:1})] Das Distributivgesetz anwenden. \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{eq6:3})] $np$ und $nq$ in den Wurzeln Kürzen und Kommutativgesetz anwenden. \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. + \item[(\ref{eq6:5})] $-z\sqrt{npq}+z\sqrt{npq}=0$ und Distributivgesetz (invers) anwenden. + \item[(\ref{eq6:6})] Es gilt $p+q=1$, da $q = 1-p$ definiert wurde. \end{itemize} - Da $e^{ln(l)} = l$ setzen wir nun $e^{-\frac{z^2}{2}}$ in \cref{eq4:3} ein kommen wir zum finalen Ergebnis: + Da $e^{ln(l)} = l$ setzen wir nun $e^{-\frac{z^2}{2}}$ in \cref{eq4:3} ein, kommen wir zum finalen Ergebnis: \begin{align} P(S_n = k) &\approx \frac{1}{\sqrt{2\pi npq}}e^{-\frac{z^2}{2}} \nonumber \\ &= \frac{1}{\sqrt{2\pi npq}}e^{-\frac{\left(\frac{k-np}{\sqrt{npq}}\right)^2}{2}} \label{eq7:1} \\ @@ -216,7 +225,7 @@ Für große $n$ wird die direkte Berechnung der Binomialverteilung sehr aufwendi wobei $\beta = \sqrt{\frac{q}{np}}$ und $\gamma = \sqrt{\frac{p}{nq}}$ definiert sind. \end{lemma} \begin{proof} - Wir betrachten exemplarisch den ersten Summanden (der zweite verhält sich vollkommen analog). Da $\beta = \sqrt{\frac{q}{np}}$, verhält sich der Restterm dritter Ordnung bezüglich $n$ wie folgt: + Wir betrachten exemplarisch den ersten Summanden (der zweite Verhält sich vollkommen analog). Da $\beta = \sqrt{\frac{q}{np}}$, verhält sich der Restterm dritter Ordnung bezüglich $n$ wie folgt: \[ O((z\beta)^3) = O\left(z^3\cdot\left(\sqrt{\frac{q}{np}}\right)^3\right) = O\left(\frac{z^3 q^{3/2}}{p^{3/2}}\cdot\frac{1}{n^{3/2}}\right) \subseteq O\left(n^{-3/2}\right) \] @@ -227,7 +236,7 @@ Für große $n$ wird die direkte Berechnung der Binomialverteilung sehr aufwendi &= C \cdot np \cdot n^{-3/2} + C \cdot |z|\sqrt{npq} \cdot n^{-3/2} \\ &= C \cdot p \cdot \frac{1}{\sqrt{n}} + C \cdot |z|\sqrt{pq} \cdot \frac{1}{n} \end{align*} - Da $C, p, q$ und $z$ von $n$ unabhängige Konstanten sind und $n$ im Nenner unbegrenzt wächst, strebt dieser Ausdruck für $n \to \infty$ gegen: + Da $C, p, q$ und $z$ in diesem Fall von $n$ unabhängige Konstanten sind und $n$ im Nenner unbegrenzt wächst, strebt dieser Ausdruck für $n \to \infty$ gegen: \[ 0 + 0 = 0 \] @@ -260,7 +269,7 @@ Das Pascalsche Dreieck ist ein geometrisches Dreieck aus Zahlen, das sich unendl p_{n,k}=\binom{n}{k} \] \end{definition} -Die ersten Zeilen des Pascalschen Dreiecks lauten +Die ersten Zeilen des Pascalschen Dreiecks lauten: \[ \begin{array}{ccccccccc} &&&&1\\ @@ -271,7 +280,7 @@ Die ersten Zeilen des Pascalschen Dreiecks lauten \end{array} \] \begin{satz}[Pascalsche Rekursion] - Für $n\geq1$ und $1\leq k\leq n-1$ gilt + Für $n\geq1$ und $1\leq k\leq n-1$ gilt: \[ \binom{n}{k} = \binom{n-1}{k-1}+\binom{n-1}{k} \] @@ -327,7 +336,7 @@ Nun haben wir nachgewiesen, dass die Funktion $\ln(1-x)$ dreimal differenzierbar \begin{bemerkung} Mit einem Blick auf die Funktion und wie ihre Ableitungen gebildet werden, ist leicht zu erkennen, dass $\ln(x-1)$ sogar unendlich oft differenzierbar ist. \end{bemerkung} -Bevor wir die Approximation nachweisen ist eine formale Definition des Satzes von Taylor und dem Taylorpolynom nötig, wobei letzteres aus dem Satz von Taylor folgt. +Bevor wir die Approximation nachweisen, ist eine formale Definition des Satzes von Taylor und dem Taylorpolynom nötig, wobei letzteres aus dem Satz von Taylor folgt. \begin{satz}[Satz von Taylor]{taylor}{\cite{enders_analysis_2024}} Seien $n\in\mathbb{N}_0, f:[a,b]\rightarrow\mathbb{R}$ eine $(n+1)$-mal differenzierbare Funktion und $x_0\in[a,b]$. @@ -342,7 +351,7 @@ Bevor wir die Approximation nachweisen ist eine formale Definition des Satzes v \end{satz} \begin{bemerkung} - Das Lagrange-Restglied ist in dein meisten Fällen eine Funktion vom Grad $n+1$, aufgrund des letzten Faktors. Es kann aber auch sein, dass das Lagrange-Restglied das Nullpolynom ist, je nach Wahl von $\xi$ und der Auswertung von der $n$-ten Ableitung von $f$ an der Stelle $\xi$. + 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 $\xi$. 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. 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