From eb79a4e05b4db6c2005b8883dd4a078e72d3f13d Mon Sep 17 00:00:00 2001 From: Luca Weidlich Date: Thu, 28 May 2026 13:07:28 +0200 Subject: [PATCH] =?UTF-8?q?ZSG=20Beweis=20Schritt=202=20und=20Anf=C3=A4nge?= =?UTF-8?q?=20von=20Schritt=203?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- content/Mathematischer_Hintergrund.tex | 64 ++++++++++++++++++++++++- galton.pdf | Bin 66975 -> 80738 bytes 2 files changed, 63 insertions(+), 1 deletion(-) diff --git a/content/Mathematischer_Hintergrund.tex b/content/Mathematischer_Hintergrund.tex index 03fbec0..240dab5 100644 --- a/content/Mathematischer_Hintergrund.tex +++ b/content/Mathematischer_Hintergrund.tex @@ -42,7 +42,7 @@ Anhand des Galton-Brettes lässt sich nun wie bereits beschrieben leicht erkenne \begin{proof} 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:} \\ + \textbf{1. Anwendung der Stirling-Formel:}\\ Wir beginnen mit der Definition der Binomialwahrscheinlichkeit: \[ P(S_n = k) = \frac{n!}{k!(n-k)!} p^k q^{n-k} @@ -60,6 +60,68 @@ Anhand des Galton-Brettes lässt sich nun wie bereits beschrieben leicht erkenne \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 + \[ + Z_n=\frac{S_n-E(X_n)}{\sigma(S_n)} = \frac{k-np}{\sqrt{npq}} + \] + definieren. Nun nimmt $Z_n$ Werte $z=\frac{k-np}{\sqrt{npq}}$ an, nach $k$ und nach $n-k$ umgestellt heißt das + \begin{align*} + k = np + x\sqrt{npq} \\ + n-k = nq - x\sqrt{npq} + \end{align*} + Nun betrachten wir $k \cdot (n-k)$ für $n \rightarrow \infty$: + \begin{align} + k(n-k) &= (np + x\sqrt{npq})\cdot(nq - x\sqrt{npq}) \label{eq3:einsetzen}\\ + &= n^2pq + (nz\sqrt{nqp}(q-p)-z^2npq) \label{eq3:umstellen}\\ + &\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: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: + \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}\\ + &= \frac{1}{\sqrt{2\pi}}\frac{1}{\sqrt{npq}} \left(\frac{np}{k}\right)^k \left(\frac{nq}{n-k}\right)^{n-k} \label{eq4:2}\\ + &= \frac{1}{\sqrt{2\pi npq}}\left(\frac{k}{np}\right)^{-k} \left(\frac{n-k}{nq}\right)^{-n+k} \label{eq4:3} + \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 + \end{itemize} + 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 Sterling-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) \\ + &= -k \ln\left(\frac{k}{np}\right) - (n-k) \ln\left(\frac{n-k}{nq}\right) \\ + &= (np + x\sqrt{npq}) \ln\left(\frac{(np + x\sqrt{npq})}{np}\right) - (nq - x\sqrt{npq}) \ln\left(\frac{(nq - x\sqrt{npq})}{nq}\right) \\ + &= (-np - x\sqrt{npq}) \ln\left(1+z\sqrt{\frac{q}{np}}\right) + (-nq + x\sqrt{npq}) \ln\left(1+z\sqrt{\frac{p}{nq}}\right) \label{eq5:3} + \end{align} + Die Taylor-Approximation besagt: $ln(1-\alpha) \approx \alpha-\frac{\alpha^2}{2}$ und ein Rest, für den wir uns in dem Fall nicht interessieren, da wir die Approximation nach dem quadratischem Term abbrechen. %wieso O(n^3) egal? + Betrachten wir nun die Logarithmen für die Tailorapproximation näher: + \begin{align*} + \ln\left(1 + x\sqrt{\frac{q}{np}}\right) &\approx x\sqrt{\frac{q}{np}} - \frac{\left(x\sqrt{\frac{q}{np}}\right)^2}{2} \\ + &= x\sqrt{\frac{q}{np}} - \frac{x^2 q}{2np} + \end{align*} + und für den anderen ln analog: + \begin{align*} + \ln\left(1 - x\sqrt{\frac{p}{nq}}\right) &\approx -x\sqrt{\frac{p}{nq}} - \frac{\left(x\sqrt{\frac{p}{nq}}\right)^2}{2} \\ + &= -x\sqrt{\frac{p}{nq}} - \frac{x^2 p}{2nq} + \end{align*} + Setzen wir dies nun zurück in \cref{eq5:3} ein, erhalten wir: + \begin{align} + &(-np - x\sqrt{npq}) \ln\left(1+z\sqrt{\frac{q}{np}}\right) + (-nq + x\sqrt{npq}) \ln\left(1+z\sqrt{\frac{p}{nq}}\right) \\ + &=(-np - x\sqrt{npq})(x\sqrt{\frac{q}{np}} - \frac{x^2 q}{2np}) + (-nq + x\sqrt{npq})(-x\sqrt{\frac{p}{nq}} - \frac{x^2 p}{2nq}) \\ + &=() + \end{align} %weiter ... \end{proof} diff --git a/galton.pdf b/galton.pdf index 689159cfab8aa20544aee695a558e88bfec9b6b9..d31b92c0e8be6b051edb8d0a87ff258814dae9fe 100644 GIT binary patch delta 50890 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