From 7033a772c6a5621e4e15dd737713721cee820f75 Mon Sep 17 00:00:00 2001 From: Luca Weidlich Date: Thu, 4 Jun 2026 17:58:03 +0200 Subject: [PATCH] Beweis weiter aber ist so falsch --- content/Mathematischer_Hintergrund.tex | 19 ++++++++++++------- galton.pdf | Bin 270539 -> 280436 bytes 2 files changed, 12 insertions(+), 7 deletions(-) diff --git a/content/Mathematischer_Hintergrund.tex b/content/Mathematischer_Hintergrund.tex index c9f5670..6f72ebc 100644 --- a/content/Mathematischer_Hintergrund.tex +++ b/content/Mathematischer_Hintergrund.tex @@ -105,23 +105,28 @@ Anhand des Galton-Brettes lässt sich nun wie bereits beschrieben leicht erkenne &= (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? + Die Taylor-Approximation besagt: $ln(1-\alpha) \approx \alpha-\frac{\alpha^2}{2}+O(x^3)$ %Hierbei beschreibt O(x^3) ... 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} + \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} + O(x^3)\\ + &= x\sqrt{\frac{q}{np}} - \frac{x^2 q}{2np} + O(x^3) \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} + \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} + O(x^3) \\ + &= -x\sqrt{\frac{p}{nq}} - \frac{x^2 p}{2nq} + O(x^3) \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}) \\ - &=() + &=(-np - x\sqrt{npq})(x\sqrt{\frac{q}{np}} - \frac{x^2 q}{2np} + O(3)) + (-nq + x\sqrt{npq}))(-x\sqrt{\frac{p}{nq}} - \frac{x^2 p}{2nq} + O(x^3)) \\ + &=(-npx\sqrt{\frac{q}{np}}+\frac{npx^2q}{2np}-x^2\sqrt{npq}\sqrt{\frac{q}{np}} + O(x^3))+ (nqx\sqrt{\frac{p}{nq}}+\frac{nqx^2p}{2nq}-x^2\sqrt{npq}\sqrt{\frac{p}{nq}} + O(x^3))\\ + &=(-x\sqrt{\frac{(np)^2q}{np}}+\frac{x^2q}{2}-x^2\sqrt{\frac{npq^2}{np}}+O(x^3)) + (x\sqrt{\frac{(nq)^2p}{nq}}+\frac{x^2p}{2}-x\sqrt{\frac{np^2q}{nq}} + O(x^3)) \\ + &=(-x\sqrt{npq}-x^2q+\frac{x^2q}{2} + O(3)) + (x\sqrt{npq}-x^2p-\frac{x^2p}{2} + O(3)) \\ + &\approx (-x\sqrt{npq}-\frac{x^2q}{2}) + (x\sqrt{npq}-\frac{x^2p}{2})\\ + &=-\frac{x^2q}{2}-\frac{x^2p}{2} \end{align} + %isso falsch! die ergebnisse kommen ins e^ 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