From 7ee020034545a934c7607f013cc05b3604fddc5b Mon Sep 17 00:00:00 2001 From: Luca Weidlich Date: Mon, 15 Jun 2026 14:16:32 +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 340621 -> 338803 bytes 2 files changed, 4 insertions(+), 3 deletions(-) diff --git a/content/Mathematischer_Hintergrund.tex b/content/Mathematischer_Hintergrund.tex index 7682926..fedfe8a 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: $ln(1-\alpha) \approx -\alpha-\frac{\alpha^2}{2}+O(\alpha^3)$ und $ln(1+\alpha) \approx \alpha-\frac{\alpha^2}{2}+O(\alpha^3)$. Hierbei steht $O(\alpha^3)$ für die Landau-Notation. Wir definieren \[ \beta := \sqrt{\frac{q}{np}}; \quad \gamma := \sqrt{\frac{p}{nq}} \] - 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 4b8090b84797e1202efa3e79dd35abf2aa574fab..ceb8a4d65f99c5fc8ef98375e646c941729ac1f6 100644 GIT binary patch delta 80822 zcmZsiV{wr$(CZQFig+d8prCnvUT+jesH-QC)%-KzaCKcK5-rux46>bMiM zr&Tm4awRbddS(XpAgC}X;8rLQv@%c@Ztix%G7vN{fC;&}UH*QVLP6XlR&z*iPs@@I zIC&}^FJhp`OV7YKbOBo-@|TBs_jqwu5R!sVCes1`A68hhigP_;pMo)bxv+&;?glp9 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