From 6e8061b94d8943f9e7084225002c8d858b282586 Mon Sep 17 00:00:00 2001 From: Luca Weidlich Date: Thu, 16 Jul 2026 11:02:45 +0200 Subject: [PATCH] luca hat noch fachliche Bredolien behoben --- bibliography.bib | 15 ++++- content/Mathematischer_Hintergrund.pdf | Bin 0 -> 25926 bytes content/Mathematischer_Hintergrund.tex | 77 +++++++++++++------------ galton.pdf | Bin 1012950 -> 1017817 bytes material/ab3.tex | 8 +-- material/ab3_Lös.pdf | Bin 210702 -> 210707 bytes material/ab3_Lös.tex | 4 +- 7 files changed, 59 insertions(+), 45 deletions(-) create mode 100644 content/Mathematischer_Hintergrund.pdf diff --git a/bibliography.bib b/bibliography.bib index 4e9f5e3..608cdd2 100644 --- a/bibliography.bib +++ b/bibliography.bib @@ -172,4 +172,17 @@ author = {Forster, Otto}, date = {2023}, note = {Num Pages: 460}, -} \ No newline at end of file +} + + +@book{georgii_stochastik_2015, + title = {Stochastik: Einführung in die Wahrscheinlichkeitstheorie und Statistik}, + isbn = {978-3-11-035969-5}, + url = {https://www.degruyter.com/document/doi/10.1515/9783110359701/html}, + doi = {10.1515/9783110359701}, + shorttitle = {Stochastik}, + publisher = {{DE} {GRUYTER}}, + author = {Georgii, Hans-Otto}, + urldate = {2026-07-16}, + date = {2015-12-31}, +} diff --git a/content/Mathematischer_Hintergrund.pdf b/content/Mathematischer_Hintergrund.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b5ac3aa91d087e6d182af368000eaeb5dc7f340e GIT binary patch literal 25926 zcmbTeQ*>=xw=KM4uGr3sZ6`Bav2EM7ZQHi(tk||~+sWPM|8KcxfA{Wl%2&?IDD!2! 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zrQ%`-Xy~y0egXvl8g-8si5vTzuT zH$==sN4TA!o2aAw);-lH@)(+6ko+I4tDRj8om@PeOwCY$oE$tnC^R%56~s~g7t|H$ AIsgCw literal 0 HcmV?d00001 diff --git a/content/Mathematischer_Hintergrund.tex b/content/Mathematischer_Hintergrund.tex index 4aff77a..1d82639 100644 --- a/content/Mathematischer_Hintergrund.tex +++ b/content/Mathematischer_Hintergrund.tex @@ -1,7 +1,7 @@ %"ltex.language": "de-DE" \subsection{Beweis des zentralen Grenzwertsatzes} -Das Galtonbrett (nach Francis Galton) dient der Veranschaulichung der Binomialverteilung und der experimentellen Bestätigung vom zentralen Grenzwertsatz im Spezialfall der Binomialverteilung. Im Folgenden formalisieren wir den Weg einer Kugel durch das Brett als stochastischen Prozess mithilfe einer Binomialverteilung und Beweisen anschließend den zentralen Grenzwertsatz von Moivre-Laplace \cite{nahrstedt_monte-carlo-methode_2015}. +Das Galtonbrett (nach Francis Galton) dient der Veranschaulichung der Binomialverteilung und der experimentellen Bestätigung vom zentralen Grenzwertsatz im Spezialfall der Binomialverteilung. Im Folgenden formalisieren wir den Weg einer Kugel durch das Brett als stochastischen Prozess mithilfe einer Binomialverteilung und beweisen anschließend den zentralen Grenzwertsatz von Moivre-Laplace \cite{nahrstedt_monte-carlo-methode_2015}. \begin{definition}[Modell des Galtonbretts]{galton}{nach \cite[S. 252-253]{buchter_elementare_2005}} Sei $(\Omega, \mathcal{F}, P)$ ein Wahrscheinlichkeitsraum. Der Fall einer Kugel durch ein Galtonbrett mit $n \in \mathbb{N}$ Reihen wird modelliert durch eine Folge von stochastisch unabhängigen und identisch verteilten Zufallsvariablen $X_1, X_2, \dots, X_n$, wobei $X_i \in \{0, 1\}$. @@ -72,28 +72,29 @@ Anschließend ist folgendes zu bemerken: Daher gilt $S_n \sim \mathcal{B}in_{n,p}$ \end{satz} \begin{proof} - Nach \cref{bem:XiB} und \cref{def:Sn} ist $S_n$ die Summe von $n$ unabhängigen Bernoulliverteilten Zufallsvariablen $X_i \sim \mathcal{B}_p$. Also stellt $S_n$ die Anzahl der Erfolge von n-Wiederholungen von unabhängigen identisch Bernoulliverteilten Zufallsexperimenten dar. + Nach \cref{bem:XiB} und \cref{def:Sn} ist $S_n$ die Summe von $n$ unabhängigen Bernoulliverteilten Zufallsvariablen $X_i \sim \mathcal{B}_p$. Also stellt $S_n$ die Anzahl der Erfolge von n-Wiederholungen von unabhängigen identisch bernoulliverteilten Zufallsexperimenten dar. Daraus folgt, dass das Galtonbrett mit der Zufallsvariable $S_n$ ein Binomialmodell nach \cref{def:binModell} darstellt und somit die Binomialverteilung nach \cref{def:binZähldichte} eine Zähldichte definiert. Daher gilt: $S_n \sim \mathcal{B}in_{n,p}$. \end{proof} Eine weitere in diesem Kontext interessante Verteilung ist die Normalverteilung. -\begin{definition}[Normalverteilung] +\begin{definition}[Normalverteilung]{norm}{\cite{kosenkova_stochastik_2025}} Sei $X$ eine stetige Zufallsvariable mit Dichte - \[ f(x)= \frac{1}{\sqrt{2\pi \sigma ^2}}\cdot e^{-\frac{\left( x-\mu \right)^2}{2\sigma ^2}}\] + \[ f_\mathcal{N(\mu, \sigma^2)}(x)= \frac{1}{\sqrt{2\pi \sigma ^2}}\cdot e^{-\frac{\left( x-\mu \right)^2}{2\sigma ^2}}\] und Parametern $\mu \in \mathbb{R}, \sigma > 0$, wobei $\mu$ den Erwartungswert und $\sigma ^2$ die Varianz von $X$ darstellt, so nennt man $X$ \emph{Normalverteilt} und schreibt: \[ X \sim \mathcal{N}(\mu, \sigma ^2)\] + Die Dichte der Normalverteilung wird auch ``Gauß’sche Glockenkurve'' genannt. \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 Wahrscheinlichkeitsverteilung von $S_n$, durch die Dichtefunktion der Normalverteilung annähern: +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 per se 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}[Lokaler Grenzwertsatz von de Moivre-Laplace]{moivrelaplace}{nach \cite{georgii_stochastik_2015}} + 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 $q = 1-p$. Dann gilt bei einem beliebigen $c>0$ mit $z=\frac{k-np}{\sqrt{npq}}$, dass \[ - S_n \sim \mathcal{N}(\mu,\sigma ^2) + \lim_{n \to \infty} \mathop{\max}\limits_{k:|z|\leq c} \left|\frac{P(S_n=k)}{f_\mathcal{N(\mu, \sigma^2)}(k)}\right| = 1 \] - wobei $\mathbb{E}(X)=\mu$, $Var(X)=\sigma ^2 < \infty$ und $\frac{k-np}{\sqrt{npq}}$ begrenzt ist. \end{satz} +In der Literatur wird allgemein vom Satz von Moivre-Laplace gesprochen, wenn die Approximation der Binomial- durch die Normalverteilung gemeint ist \cite{statistik_verstehen_beweis_2019}. Mathematisch präzise muss jedoch zwischen dem globalen und dem lokalen Grenzwertsatz unterschieden werden \cite{georgii_stochastik_2015, kosenkova_stochastik_2025}. Wir beweisen im Folgenden den lokalen Grenzwertsatz, der das (lokale) Annähern des ``oberen Randes'' des Histogramms der Binomialverteilung an die Glockenkurve beschreibt \cite{georgii_stochastik_2015}. Hierbei wird \emph{nicht} die Approximation der Binomial durch die Normalverteilung bewiesen. \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 folgt zum großen Teil dem Beweis von \cite{statistik_verstehen_beweis_2019} und wurde von den Autoren aufgrund der Konkretisierung des Satzes \cite{georgii_stochastik_2015} 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.\\ - Sei $k \in \{x| x \in\mathbb{N}_0 \land x \leq n\}$ und $\frac{k-np}{\sqrt{npq}}$ begrenzt sowie $\mathbb{E}(X)=\mu$, $Var(X)=\sigma ^2 < \infty$.\\ + Sei $k \in \{0,1,\dots,n\}$ und $\frac{k-np}{\sqrt{npq}}$ durch $c$ begrenzt sowie $\mathbb{E}(S_n)=\mu$, $Var(S_n)=\sigma ^2 < \infty$.\\ \textbf{1. Anwendung der Stirling-Formel:} \\ Wir beginnen mit der Definition der Binomialwahrscheinlichkeit: @@ -115,7 +116,7 @@ Für große $n$ wird die direkte Berechnung der Binomialverteilung aufgrund der \end{itemize} \textbf{2. Substitution und Approximation der Wurzeln:} \\ In diesem Teil des Beweises werden O-Notationen verwendet und ein grundlegendes Verständnis dieser vorausgesetzt.\\ - Um das Breiter werden der Verteilung und das Abwandern des Erwartungswertes zu verhindern standardisieren wir die Zufallsvariable, indem wir + Um das Breiterwerden der Verteilung und das Abwandern des Erwartungswertes zu verhindern standardisieren wir die Zufallsvariable, indem wir \[ Z_n=\frac{S_n-E(S_n)}{\sigma(S_n)} \] @@ -149,7 +150,7 @@ Für große $n$ wird die direkte Berechnung der Binomialverteilung aufgrund der Im Folgenden werden die nummerierten Schritte der Gleichungsketten erläutert: \begin{itemize} \item[(\ref{eq4:1})] Die Approximation verwenden. - \item[(\ref{eq4:2})] $n$ Kürzen und die Potenz-/Wurzelgesetze anwenden, um den Bruch zu vereinfachen. + \item[(\ref{eq4:2})] $n$ kürzen und die Potenz-/Wurzelgesetze anwenden, um den Bruch zu vereinfachen. \item[(\ref{eq4:3})] Den Kehrwert der Brüche bilden und die Wurzeln zusammenfassen. \end{itemize} Es ist bereits der korrekte konstante Faktor der gaußschen Glockenkurve erkennbar.\\ @@ -193,12 +194,12 @@ Für große $n$ wird die direkte Berechnung der Binomialverteilung aufgrund der \ln\left(1 - z\sqrt{\frac{p}{nq}}\right) &\sim -z\sqrt{\frac{p}{nq}} - \frac{\left(z\sqrt{\frac{p}{nq}}\right)^2}{2} + O((-z\gamma)^3) \\ &= -z\sqrt{\frac{p}{nq}} - \frac{z^2 p}{2nq} + O((z\gamma)^3) &&\text{"-" fällt wegen O weg} \end{align*} - Setzen wir dies nun zurück in \cref{eq5:3} ein, erhalten wir: + 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)+O(z^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)) + O(z^3) \nonumber \\ - &=(-npz\sqrt{\frac{q}{np}}+\frac{npz^2q}{2np}-z^2\sqrt{npq}\sqrt{\frac{q}{np}} + O((z\beta)^3)) + O(z^3) \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}\\ - &=(-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}\\ + &(-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 \\ + &=(-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} \\ &\sim (-z\sqrt{npq}-\frac{z^2q}{2}) + (z\sqrt{npq}-\frac{z^2p}{2})\label{eq6:4}\\ &=-\frac{z^2q}{2}-\frac{z^2p}{2} = -\frac{z^2}{2}(p+q) \label{eq6:5}\\ @@ -206,10 +207,10 @@ Für große $n$ wird die direkte Berechnung der Binomialverteilung aufgrund der \end{align} %(-np - z\sqrt{npq}) \cdot O((z\beta)^3) + (-nq + z\sqrt{npq}) \cdot O((z\gamma)^3) Im Folgenden werden die nummerierten Schritte der Gleichungsketten erläutert: \begin{itemize} - \item[(\ref{eq6:1})] Das Distributivgesetz anwenden. - \item[(\ref{eq6:2})] $np$ und $nq$ mithilfe des Quadrates in die Wurzel ziehen, sowie Kürzen und Wurzeln zusammenfassen. Alle Terme mit $z^3$ werden von $O(z^3)$ nach dessen Definition "absorbiert". - \item[(\ref{eq6:3})] $np$ und $nq$ in den Wurzeln Kürzen und Kommutativgesetz anwenden. - \item[(\ref{eq6:4})] Die Landau Terme können vernachlässigt werden, da diese Terme für $n\rightarrow\infty$ gegen 0 gehen, sofern $z$ beschränkt ist. Betrachte Hierzu \cref{lem:restterme} und \cref{prop:restterme2}. + \item[(\ref{eq6:1})] Das Distributivgesetz anwenden. Alle Terme die zu $O\left(\frac{1}{\sqrt{n}}\right)$ passen, werden nach dessen Definition "absorbiert". + \item[(\ref{eq6:2})] $np$ und $nq$ mithilfe des Quadrates in die Wurzel ziehen, sowie Kürzen und Wurzeln zusammenfassen. Außerdem gilt trivialerweise, dass $O\left(\frac{1}{\sqrt{n}}\right) \xrightarrow{n \rightarrow \infty} 0$ + \item[(\ref{eq6:3})] $np$ und $nq$ in den Wurzeln Kürzen und Kommutativgesetz anwenden. + \item[(\ref{eq6:4})] Die Landau Terme können vernachlässigt werden, da diese Terme für $n\rightarrow\infty$ gegen 0 gehen, da $z$ beschränkt ist. Betrachte Hierzu \cref{lem:restterme} und \cref{prop:restterme2}. \item[(\ref{eq6:5})] $-z\sqrt{npq}+z\sqrt{npq}=0$ und Distributivgesetz (invers) anwenden. \item[(\ref{eq6:6})] Es gilt $p+q=1$, da $q = 1-p$ definiert wurde. \end{itemize} @@ -218,52 +219,52 @@ 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} \\ - &= \frac{1}{\sqrt{2\pi \sigma^2}}e^{\left(-\frac{(k-\mu)^2}{2\sigma^2}\right)} \label{eq7:3} + &= \frac{1}{\sqrt{2\pi \sigma^2}}e^{\left(-\frac{(k-\mu)^2}{2\sigma^2}\right)} \label{eq7:3} \\ + &= f_{\mathcal{N}(\mu, \sigma^2)}(k) \label{eq7:4} \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 von $\mu$ und $\sigma$ angewendet. + \item[(\ref{eq7:4})] Definition der Dichtefunktion angewendet. \end{itemize} - Daher konvergiert die Zähldichte einer binomialverteilte Zufallsvariable (für $n\rightarrow\infty$) gegen die Dichtefunktion der Normalverteilung. + Da wir hiermit gezeigt haben, dass $P(S_n = k) \sim f_{\mathcal{N}(\mu, \sigma^2)}(k)$ für alle $k$ gilt, deren standardisierte Abweichung $z$ beschränkt bleibt ($|z| \le c$), folgt daraus direkt die asymptotische Gleichheit. Das bedeutet für den Quotienten aus exakter Wahrscheinlichkeit und approximierter Dichte, dass er im betrachteten Intervall gleichmäßig gegen $1$ konvergiert.\\ + Zusammenfassend lässt sich festhalten: Die Zähldichte einer binomialverteilten Zufallsvariable nähert sich für $n \to \infty$ im Bereich um den Erwartungswert punktweise der Dichtefunktion der Normalverteilung an. \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. Für die Restterme der Taylor-Entwicklung des Logarithmus multipliziert mit ihren jeweiligen Vorfaktoren gilt für $n \rightarrow \infty$: + Seien $p \in (0,1)$, $q = 1-p$ und $z=\frac{k-np}{\sqrt{npq}}$ Werte einer standardisierten, binomialverteilten Zufallsvariable wobei $z$ durch ein $c$ begrenzt ist. Für die Restterme der Taylor-Entwicklung des Logarithmus multipliziert mit ihren jeweiligen Vorfaktoren gilt für $n \rightarrow \infty$: \[ (-np - z\sqrt{npq}) \cdot O((z\beta)^3) + (-nq + z\sqrt{npq}) \cdot O((z\gamma)^3) \xrightarrow{n \rightarrow \infty} 0 \] - wobei $\beta = \sqrt{\frac{q}{np}}$ und $\gamma = \sqrt{\frac{p}{nq}}$ definiert sind. + wobei $\beta := \sqrt{\frac{q}{np}}$ und $\gamma := \sqrt{\frac{p}{nq}}$ definiert sind. \end{lemma} \begin{proof} - Da $z$ begrenzt ist, gilt: $ \exists M \in \mathbb{N}: |z| 0$, sodass der Betrag dieses Restterms durch $C \cdot n^{-3/2}$ nach oben beschränkt ist. + Nach der Definition der Landau-Notation existiert für hinreichend große $n$ eine Konstante $M > 0$, sodass der Betrag dieses Restterms durch $M \cdot n^{-3/2}$ nach oben beschränkt ist. Multiplizieren wir dies mit dem Betrag des Vorfaktors, erhalten wir mithilfe der Dreiecksungleichung: \begin{align*} - \left| (-np - z\sqrt{npq}) \cdot O(n^{-3/2}) \right| &\leq (np + |z|\sqrt{npq}) \cdot C \cdot n^{-3/2} \\ - &= C \cdot np \cdot n^{-3/2} + C \cdot |z|\sqrt{npq} \cdot n^{-3/2} \\ - &= C \cdot p \cdot \frac{1}{\sqrt{n}} + C \cdot |z|\sqrt{pq} \cdot \frac{1}{n} + \left| (-np - z\sqrt{npq}) \cdot O(n^{-3/2}) \right| &\leq (np + |z|\sqrt{npq}) \cdot M \cdot n^{-3/2} \\ + &= M \cdot np \cdot n^{-3/2} + M \cdot |z|\sqrt{npq} \cdot n^{-3/2} \\ + &= M \cdot p \cdot \frac{1}{\sqrt{n}} + M \cdot |z|\sqrt{pq} \cdot \frac{1}{n} \end{align*} - Da $C, p, q$ und $z$ in diesem Fall von $n$ unabhängige Konstanten sind und $n$ im Nenner unbegrenzt wächst, strebt dieser Ausdruck für $n \to \infty$ gegen: + Da $M, p, q$ und $z$ in diesem Fall von $n$ unabhängige Konstanten sind und $n$ im Nenner unbegrenzt wächst, strebt dieser Ausdruck für $n \to \infty$ gegen: \[ 0 + 0 = 0 \] Für den zweiten Term mit $\gamma$ erfolgt der Beweis völlig analog, womit die Summe beider Terme ebenfalls gegen $0$ konvergiert. \end{proof} -Folgende Proposition ist leicht aus dem oberen Beweis ableitbar (aufgrund der Abschätzung mit $n^{-3/2}$ und der Betrachtung von z als beschränkt) und wird daher nicht separat bewiesen. +Folgende Proposition ist leicht aus dem oberen Beweis ableitbar (aufgrund der Abschätzung mit $n^{-3/2}$ und der Betrachtung von $z$ als beschränkt) und wird daher nicht separat bewiesen. \begin{proposition}[Asymptotisches Verschwinden der Taylor-Restterme ohne Präfix]{restterme2} - Seien $p \in (0,1)$, $q = 1-p$ und $z$ Werte einer standardisierten, binomialverteilten Zufallsvariable wobei $z$ begrenzt ist. Für die Restterme der Taylor-Entwicklung des Logarithmus gilt für $n \rightarrow \infty$: + Seien $p \in (0,1)$, $q = 1-p$ und $z$ Werte einer standardisierten, binomialverteilten Zufallsvariable wobei $z$ durch ein $c$ begrenzt ist. Für die Restterme der Taylor-Entwicklung des Logarithmus gilt für $n \rightarrow \infty$: \[ O((z\beta)^3) + O((z\gamma)^3) \xrightarrow{n \rightarrow \infty} 0 \] - \[ - O(z^3) \xrightarrow{n \rightarrow \infty} 0 - \] wobei $\beta = \sqrt{\frac{q}{np}}$ und $\gamma = \sqrt{\frac{p}{nq}}$ definiert sind. \end{proposition} diff --git a/galton.pdf b/galton.pdf index 78417edcd2b9d56e2127a90ac03b5f02aaee9d68..afbd17295fa09a7cc9604fb16067e3994e001f14 100644 GIT binary patch delta 156497 zcmY(KQ*$MX7KCHlJGN~*nb@{%duPIlZQHgdwr$&X=API4@cn{SwW_gHl@Y!BzPag96fK;L$sTX#l;lSzN`R~VYC0d?1|J8OS%DvK(Q z5bX3HvT=gCS;C@@&k>FX3QUmBbjeFk+b(C*`GaXIj4pz9Nw@!xj)9+H0MP$+*HOrR z*Uyp@9TVraRdF^8nU<<8M^tS-m8&gpR%Z~qW*TT&1J0jQ-EbV!)6B(~(xz?eIPq9> z)vUNT=`&}J+}sj>kFmFjckOfHyO;4|9e}Rt8~k};EkL{uD>(Q!$BR}pc>AVkQ8?Xl z>+xx_j0mpZgH6)>78EvC2aKTjt45CvBy)CJMwo9BGLQGR1HUf|I z8RA}U>xP{8_ZS~6{yW9HR$f6KsRwaFw9Zk|(7i zCUu#H%N;84s6njpIIex!S*c;Bu4p!W7EG_cO4nTtqGGKIM 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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: + Zeichne in dem folgenden Koordinatensystem, in dem auch die Gammafunktion (für $\Gamma(x+1)$) abgebildet ist, die Funktionswerte der Fakultät (welche nur auf $\mathbb{N}$ definiert ist) für $x\in \{0,1,2,3,4,5\}$ ein: \begin{figure}[H] \centering \begin{tikzpicture} @@ -328,13 +328,13 @@ thick ] {sqrt(2*pi)*((x+1/2)^(x+1/2))*exp(-(x+1/2))}; - \addlegendentry{$\Gamma(x)$} + \addlegendentry{$\Gamma(x+1)$} \end{axis} \end{tikzpicture} \end{figure} Mithilfe dieser Funktion lässt sich die Fakultät leicht approximieren. Diese Approximation nennt sich Sterling Approximation und besagt: \[ - n! \sim \sqrt{2\pi n} \left(\frac{n}{e}\right)^n \sim \Gamma(n) + n! \sim \sqrt{2\pi n} \left(\frac{n}{e}\right)^n \sim \Gamma(n+1) \] %Stimmt das mit dem Gamma? Setzen wir das in die Gleichung für $P(S_n = k)$ ein, erhalten wir nach umstellen: \[ @@ -342,7 +342,7 @@ \] \item Aus dieser Formel lässt sich nun leicht schon einmal der Vorfaktor der Normalverteilung erschließen. Hierzu und zu Vereinfachung weiterer Rechnungen standardisieren wir $S_n$. Dies entspricht auch einer einfachen Substitution und daher vollkommen legitim. Hierzu definieren wir: \[ - Z_n = Z_n=\frac{S_n-E(S_n)}{\sigma(S_n)} + Z_n=\frac{S_n-E(S_n)}{\sigma(S_n)} \] nimmt $Z_n$ die folgenden Werte an: \[ diff --git a/material/ab3_Lös.pdf b/material/ab3_Lös.pdf index 217495f180cac5ece31041587462d9c8720c3caf..a85dcff46fa68388b7f3a21d4142b9d0ce58a340 100644 GIT binary patch delta 10954 zcmai4Q&c97+f18nZnkaPwr$tu-s;V6v)kUV+16&;wr$t{clq6Y_s`75oH=vO%z5Ik z5R_S5D?jCNIrjGF5YnO)O@suq|o737wUYocZ9NQ=g%FIwJ5S_RR z`Vfe*hhgP%jF9ATPL4zO&+bZ4qZUV#`UzJ?uyMf~)#a@UY`C$*Q6$M;ueB?KJHj0n zEe9IQYU%I!qA&R0ft_!!i!I~V-!|#(O9yjzm+ymbi!befmDZt?bZ6qPx1o%iF4m{5 z42`(k>)YwzUoEo`-%8HkPnY@F*>_p;QrmAfQ#NL|BhSvvTOCWfjn0NgrBcL7C7xqF z4ZjF)rD@(eI`)s*f*PDSozC<7zk`+}0m`BPWv@%;4+Yh{kNy_!;u%1oj+U{!!IyPp zZ#8Y|;V~uPrJgUJZ@2U?`*5=%%*XWwt|awPV+$KT1Y=5+^29&+2=c3q%66IJog~?L z@4U*NY7n?L6zvfW+x57xOoKyI9Fx+h@s3dx`ULfkU-2R0`g72WQyE9wy5H@V#I&1l z`XukMa@rp0^rq6@-vSg35mtwp3*Jxb8eMI#REh{-6-exh`0KU)AkF{%v?rI|Gym05 z`ToJ{JRI-l-4H-_C)X9=CT+E0cz*e*kN0OiyeHd_93)0Gz5cYJ^V#h5<>WMorGT=} 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Hierzu und zu Vereinfachung weiterer Rechnungen standardisieren wir $S_n$. Dies entspricht auch einer einfachen Substitution und daher vollkommen legitim. Hierzu definieren wir: \[ - Z_n = Z_n=\frac{S_n-E(S_n)}{\sigma(S_n)} + Z_n=\frac{S_n-E(S_n)}{\sigma(S_n)} \] nimmt $Z_n$ die folgenden Werte an: \[