From 1d3db29ac3f5ea9882433094ce5cddff30a75f04 Mon Sep 17 00:00:00 2001 From: Luca Weidlich Date: Mon, 15 Jun 2026 13:52:13 +0200 Subject: [PATCH] =?UTF-8?q?kleiner=20Vorzeichenfehler=20korrigiert=20und?= =?UTF-8?q?=20Fall=20eingef=C3=BCgt?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- content/Mathematischer_Hintergrund.tex | 7 ++++--- galton.pdf | Bin 339146 -> 338403 bytes 2 files changed, 4 insertions(+), 3 deletions(-) diff --git a/content/Mathematischer_Hintergrund.tex b/content/Mathematischer_Hintergrund.tex index c9bb954..a24b2cc 100644 --- a/content/Mathematischer_Hintergrund.tex +++ b/content/Mathematischer_Hintergrund.tex @@ -151,16 +151,17 @@ Für große $n$ wird die direkte Berechnung der Binomialverteilung sehr aufwendi \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: $ln(1-\alpha) \approx \alpha-\frac{\alpha^2}{2}+O(\alpha^3)$ Hierbei steht $O(\alpha^3)$ für die Landau-Notation. + \cref{def:taylorpolynom} beweist die Taylor-Approximation für den Logarithmus: $ln(1-\alpha) \approx -\alpha-\frac{\alpha^2}{2}+O(\alpha^3)$ sowie $ln(1+\alpha) \approx \alpha-\frac{\alpha^2}{2}+O(\alpha^3)$ 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) Hierbei genügt die Abschätzung gegen unendlich, da wir die Annäherung nur für große Zahlen beweisen wollen.\\ + Um diese Approximation zu verwenden, muss sicher sein, dass $-1 < |\alpha| < 1$. Das heißt, dass $\forall z \in \mathbb{R}: -1 \overset{n\rightarrow\infty}{<} |z\cdot\beta| \overset{n\rightarrow\infty}{<}1$. (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 + |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\\ + und \quad -1 <0 <1 \end{align*} Da $p$ und $q$ konstante Parameter sind. diff --git a/galton.pdf b/galton.pdf index 2bbdbeeadaead6749a2faa0d935e22d1714b4b46..11392df2501ca49f965b54d703743319fec23baa 100644 GIT binary patch delta 68885 zcmV(~K+nI**%af`6o7;QgaWh!A3A@At(Sh6vt@)BN6Mn>u4>daQK97N4xUYdKu^Zn zw$+3Qa)R`GLktKJU(r5EZ=^X+ds?aGN<0FTB>WApC*64!C4TYUitg64)fFT+!82`P zq6*IE5E{YIqs1va8e%Y8LbAgnr#N4maim~(CEOzvD<|@!-F5W!LC)hk9&3NR$8*JF 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