From 7cc5c24d345c566ae5a58bee81386250c723099c Mon Sep 17 00:00:00 2001 From: Luca Weidlich Date: Sat, 18 Jul 2026 09:15:01 +0200 Subject: [PATCH] klammer zu viel entfernt --- content/Mathematischer_Hintergrund.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/content/Mathematischer_Hintergrund.tex b/content/Mathematischer_Hintergrund.tex index e626046..ba0f5bf 100644 --- a/content/Mathematischer_Hintergrund.tex +++ b/content/Mathematischer_Hintergrund.tex @@ -197,7 +197,7 @@ In der Literatur wird allgemein vom Satz von Moivre-Laplace gesprochen, wenn die Setzen wir dies nun zurück in \cref{eq5:4} 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 \\ + &=(-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)) + O\left(\frac{1}{\sqrt{n}}\right) \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}\\ &\sim(-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} \\