From 781f305a14657042dfa636a6651a50534996c817 Mon Sep 17 00:00:00 2001 From: Luca Weidlich Date: Wed, 15 Jul 2026 15:45:07 +0200 Subject: [PATCH] =?UTF-8?q?Kleine=20Fachliche=20Pr=C3=A4zisierung?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- content/Mathematischer_Hintergrund.tex | 13 +++++++------ galton.pdf | Bin 1321702 -> 1012950 bytes material/ab3.pdf | Bin 302585 -> 146831 bytes material/ab3.tex | 8 ++++---- material/ab3_Lös.pdf | Bin 369801 -> 210702 bytes material/ab3_Lös.tex | 8 ++++---- 6 files changed, 15 insertions(+), 14 deletions(-) diff --git a/content/Mathematischer_Hintergrund.tex b/content/Mathematischer_Hintergrund.tex index 3d922ca..4aff77a 100644 --- a/content/Mathematischer_Hintergrund.tex +++ b/content/Mathematischer_Hintergrund.tex @@ -84,15 +84,16 @@ Eine weitere in diesem Kontext interessante Verteilung ist die Normalverteilung. \end{definition} Für große $n$ wird die direkte Berechnung der Binomialverteilung aufgrund der Fakultäten sehr aufwendig. An dieser Stelle greift der zentrale Grenzwertsatz, der besagt, dass sich die Binomialverteilung für große $n$ der Normalverteilung annähert. Für große $n$ lässt sich damit die Wahrscheinlichkeit eines Ereignisses (aufgrund der Standardisierung) deutlich leichter bestimmen. Des Weiteren ist es mathematisch gesehen sehr schön zu sehen, wie eine relativ einfache diskrete Verteilung, wie die Binomialverteilung, mit einer mathematisch persé sehr anders aussehenden Verteilung zusammenhängt. Im Spezialfall der Binomialverteilung wird dies durch den Satz von Moivre-Laplace formalisiert \cite{nahrstedt_monte-carlo-methode_2015, statistik_verstehen_beweis_2019}. \begin{satz}[Zentraler Grenzwertsatz von de Moivre-Laplace]{moivrelaplace}{nach \cite{kosenkova_stochastik_2025}} - Sei $S_n \sim \mathcal{B}in_{n,p}$ die Anzahl der Erfolge bei $n$ unabhängigen Bernoulli-Versuchen mit Erfolgswahrscheinlichkeit $p \in (0, 1)$ und sei $q = 1-p$. Für große $n$ lässt sich die Wahrscheinlichkeit, dass $S_n$ den Wert $k \in \{x| x \in\mathbb{N}_0 \land x \leq n\}$ annimmt, durch die Dichtefunktion der Normalverteilung annähern: + Sei $S_n \sim \mathcal{B}in_{n,p}$ die Anzahl der Erfolge bei $n$ unabhängigen Bernoulli-Versuchen mit Erfolgswahrscheinlichkeit $p \in (0, 1)$ und sei $q = 1-p$. Für große $n$ lässt sich die Wahrscheinlichkeitsverteilung von $S_n$, durch die Dichtefunktion der Normalverteilung annähern: \[ - P(S_n = k) \sim \mathcal{N}(\mu,\sigma ^2) + S_n \sim \mathcal{N}(\mu,\sigma ^2) \] wobei $\mathbb{E}(X)=\mu$, $Var(X)=\sigma ^2 < \infty$ und $\frac{k-np}{\sqrt{npq}}$ begrenzt ist. \end{satz} \begin{proof} Der Beweis folgt zum großen Teil dem Beweis von \cite{statistik_verstehen_beweis_2019} und wurde von den Autoren weiter konkretisiert.\\ - 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 drei Approximationen werden in diesem Beweis als wahr angenommen, jedoch in der folgenden Arbeit weiter analysiert und bewiesen. + 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 drei Approximationen werden in diesem Beweis als wahr angenommen, jedoch in der folgenden Arbeit weiter analysiert und bewiesen.\\ + Sei $k \in \{x| x \in\mathbb{N}_0 \land x \leq n\}$ und $\frac{k-np}{\sqrt{npq}}$ begrenzt sowie $\mathbb{E}(X)=\mu$, $Var(X)=\sigma ^2 < \infty$.\\ \textbf{1. Anwendung der Stirling-Formel:} \\ Wir beginnen mit der Definition der Binomialwahrscheinlichkeit: @@ -217,15 +218,15 @@ Für große $n$ wird die direkte Berechnung der Binomialverteilung aufgrund der P(S_n = k) &\sim \frac{1}{\sqrt{2\pi npq}}e^{-\frac{z^2}{2}} \nonumber \\ &= \frac{1}{\sqrt{2\pi npq}}e^{-\frac{\left(\frac{k-np}{\sqrt{npq}}\right)^2}{2}} \label{eq7:1} \\ &= \frac{1}{\sqrt{2\pi npq}}e^{\left(-\frac{(k-np)^2}{2npq}\right)} \label{eq7:2} \\ - &= \mathcal{N}(np)(npq) = \mathcal{N}(\mu)(\sigma^2) \label{eq7:3} + &= \frac{1}{\sqrt{2\pi \sigma^2}}e^{\left(-\frac{(k-\mu)^2}{2\sigma^2}\right)} \label{eq7:3} \end{align} Im Folgenden werden die nummerierten Schritte der Gleichungsketten erläutert: \begin{itemize} \item[(\ref{eq7:1})] Rücksubstitution der Standardisierung. \item[(\ref{eq7:2})] Der Bruch wurde quadriert und zusammengefasst. - \item[(\ref{eq7:3})] Definition der Standardnormalverteilung angewendet. + \item[(\ref{eq7:3})] Definition von $\mu$ und $\sigma$ angewendet. \end{itemize} - Daher konvergiert eine binomialverteilte Zufallsvariable (für $n\rightarrow\infty$) gegen die Dichtefunktion der Normalverteilung. + Daher konvergiert die Zähldichte einer binomialverteilte Zufallsvariable (für $n\rightarrow\infty$) gegen die Dichtefunktion der Normalverteilung. \end{proof} \begin{lemma}[Asymptotisches Verschwinden der Taylor-Restterme mit Präfix]{restterme} Seien $p \in (0,1)$, $q = 1-p$ und $z$ Werte einer standardisierten, binomialverteilten Zufallsvariable wobei z begrenzt ist. 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Hierzu formulieren wir dies zuerst mathematisch formal sauber: \begin{satzbox}[Zentraler Grenzwertsatz von de Movire-Laplace] Sei $S_n \sim \mathcal{B}in_{n,p}$ die Anzahl der Erfolge bei $n$ unabhängigen Bernoulli-Versuchen mit Erfolgswahrscheinlichkeit $p \in (0, 1)$ und sei $q = 1-p$ die Gegenwahrscheinlichkeit. (Beispielsweise bei einem Galtonbrett)\\ - Für große $n$ lässt sich die Wahrscheinlichkeit, dass $S_n$ den Wert $k \in \{x| x \in\mathbb{N}_0 \land x \leq n\}$ annimmt, durch die Dichtefunktion der Normalverteilung annähern: + Für große $n$ lässt sich die Zähldichte von $S_n$ durch die Dichtefunktion der Normalverteilung annähern: \[ - P(S_n = k) \sim \mathcal{N}(\mu,\sigma ^2) + S_n \sim \mathcal{N}(\mu,\sigma ^2) \] wobei $\mathbb{E}(X)=\mu$, $Var(X)=\sigma ^2$ und $\frac{k-np}{\sqrt{npq}}$ begrenzt ist. \end{satzbox} @@ -303,7 +303,7 @@ \answer{3} \item Zuerst muss uns nochmal klar werden, was unser Ausgangspunkt und und das Ziel ist:\\ Ausgangspunkt: $P(S_n = k) = $ \antwortbox[6cm] \vspace{1em} \\ - Ziel: $\mathcal{N}(\mu,\sigma ^2) = \mathcal{N}(np)(npq) = $ \antwortbox[6cm] + Ziel: $\frac{1}{\sqrt{2\pi \sigma ^2}}\cdot e^{-\frac{\left( x-\mu \right)^2}{2\sigma ^2}}=$ \antwortbox[6cm] \item Wir wollen die Binomialverteilung als diskrete Verteilung in eine stetige Verteilung überführen. Das Aspekt der Binomialverteilung welcher die Diskretheit forciert, ist die Fakultät. Die Fakultät der Reelen Zahlen ist die so genannte Gamma ($\Gamma$) Funktion. \\ Zeichne in dem folgenden Koordinatensystem, in dem auch die Gammafunktion abgebildet ist, die Funktionswerte der Fakultät (welche nur auf $\mathbb{N}$ definiert ist) für $x\in \{0,1,2,3,4,5\}$ ein: @@ -405,7 +405,7 @@ &= \antwortbox[8cm] \vspace{1em} \\ &= \antwortbox[8cm] \vspace{1em} \\ &= \antwortbox[8cm] \vspace{1em} \\ - &= \mathcal{N}(np)(npq) = \mathcal{N}(\mu)(\sigma^2) + &= \frac{1}{\sqrt{2\pi \sigma ^2}}\cdot e^{-\frac{\left( x-\mu \right)^2}{2\sigma ^2}} \end{align*} Daher konvergiert eine Binomialverteilte Zufallsvariable (für $n\rightarrow\infty$) gegen die Dichtefunktion der Normalverteilung. \end{enumerate} diff --git a/material/ab3_Lös.pdf b/material/ab3_Lös.pdf index 8d037015190885e9785ea4941b65b44b48b10c63..217495f180cac5ece31041587462d9c8720c3caf 100644 GIT binary patch literal 210702 zcmb@t1yq#ZyDyF)D2=2vA}HP64FXCF5<_?Q&>{i?0-`k1BF&IeLr5y!NDD}JH{3n= 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