From ce4ab1fa083216686484d59e8c7bcdc53029b2d9 Mon Sep 17 00:00:00 2001 From: Luca Weidlich Date: Wed, 1 Jul 2026 13:38:04 +0200 Subject: [PATCH] =?UTF-8?q?approx=20zu=20sim=20und=20Absch=C3=A4tzungszeug?= =?UTF-8?q?s=20f=C3=BCr=20MovireLaplace=20konkretisieren?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- bibliography.bib | 1 - content/Mathematischer_Hintergrund.tex | 86 ++++++++++++++----------- galton.pdf | Bin 446741 -> 461075 bytes 3 files changed, 48 insertions(+), 39 deletions(-) diff --git a/bibliography.bib b/bibliography.bib index 6947423..d3819a7 100644 --- a/bibliography.bib +++ b/bibliography.bib @@ -33,7 +33,6 @@ @collection{buchter_elementare_2005, location = {Berlin, Heidelberg}, title = {Elementare Stochastik: Eine Einführung in die Mathematik der Daten und des Zufalls}, - isbn = {978-3-540-22250-7 978-3-540-27368-4}, doi = {10.1007/b138982}, series = {Mathematik für das Lehramt}, shorttitle = {Elementare Stochastik}, diff --git a/content/Mathematischer_Hintergrund.tex b/content/Mathematischer_Hintergrund.tex index b13ce2f..7c19a5c 100644 --- a/content/Mathematischer_Hintergrund.tex +++ b/content/Mathematischer_Hintergrund.tex @@ -2,7 +2,7 @@ \subsection{Beweis des zentralen Grenzwertsatzes} Das Galton-Brett (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 Galton Brett]{galton}{nach \cite[S. 252-253]{buchter_elementare_2005}} - Sei $(\Omega, \mathcal{F}, P)$ ein Wahrscheinlichkeitsraum. \\ + Sei $(\Omega, \mathcal{F}, P)$ ein Wahrscheinlichkeitsraum. Der Fall einer Kugel durch ein Galton-Brett 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 $X_i \in \{0, 1\}$. Dabei beschreibt \textbf{$X_i = 1$ den Fall nach rechts} in der $i$-ten Reihe und \textbf{$X_i = 0$ den Fall nach links}. Die Wahrscheinlichkeit sei $P(X_i = 1) = p$ und $P(X_i = 0) = 1-p = q$. Bei einem symmetrischen Brett gilt $p = q = 0.5$. \end{definition} %Bild? @@ -24,7 +24,10 @@ Das Galton-Brett (nach Francis Galton) dient der Veranschaulichung der Binomialv \label{fig:galtonXi} \end{minipage} \end{figure} -Hierbei ist anzumerken, dass jedes $X_i$ Bernoulliverteilt ist, da +\begin{bemerkung}[$X_i$ Bernoulliverteilt]{XiB} + Hierbei ist anzumerken, dass jedes $X_i$ Bernoulliverteilt ist. +\end{bemerkung} +Dies gilt, da \[ \Omega = \{1,0\} \quad \text{und} \quad P(X_i = x) = @@ -33,7 +36,7 @@ Hierbei ist anzumerken, dass jedes $X_i$ Bernoulliverteilt ist, da 1-p & \text{wenn } x=0 \end{cases} \] -Somit lässt sich auch $X_i \sim \mathcal{B}_{0,5}$ schreiben.\\ +Somit lässt sich auch $X_i \sim \mathcal{B}_{p}$ schreiben.\\ Um jeden Ausgang des Galton-Brettes durchnummeriert von links nach rechts unterscheiden zu können, definieren wir uns eine weitere Zufallsvariable $S_n$ wie folgt: \begin{definition}[Zufallsvariable $S_n$]{Sn}{nach \cite[min. 0:50]{statistik_verstehen_beweis_2019}} Die Endposition der Kugel im Fach $k \in \{0, 1, \dots, n\}$ wird durch die Summe der Rechtsabbiegungen beschrieben. Wir definieren die Zufallsvariable: \[S_n = \sum_{i=1}^n X_i\] @@ -59,29 +62,32 @@ Betrachten wir nun den Binomialkoeffizienten in Bezug auf $S_n$, beschreibt dies eine Zähldichte. \end{definition} Im Fall des Galtonbrettes können wir $\Sigma = S_n$ wählen. -\begin{definition}{binZähldichte}{\cite{kosenkova_stochastik_2025}} +\begin{definition}[Binomialverteilung]{binZähldichte}{\cite{kosenkova_stochastik_2025}} Die Wahrscheinlichkeitsverteilung mit der Zähldichte $Bin_{n,p}(\{k\})$ auf $\{0,...,n\}$ heißt Binomialverteilung zu den Parametern $n, p$. \end{definition} Anschließend ist folgendes zu bemerken: -\begin{satz}[Verteilung der Endposition]{satz:binomialverteilung} +\begin{satz}[Verteilung der Endposition]{binomialverteilung} Die Wahrscheinlichkeitsverteilung der Zufallsvariable $S_n$, lässt sich durch die Binomialverteilung beschreiben.\\ Daher gilt $S_n \sim \mathcal{B}in_{n,p}$ \end{satz} \begin{proof} - Nach \cref{def:galton} und \cref{def:Sn} ist $S_n$ die Summe von $n$ unabhängigen Bernoulli-verteilten Zufallsvariablen $X_i \sim \mathcal{B}_p$. Also stellt $S_n$ die Anzahl der Erfolge von n-Wiederholungen von unabhängigen identisch Bernoulli-verteilten Zufallsexperimenten dar. - Daraus folgt, dass das Galton-Brett mit der Zufallsvariable $S_n$ eine Binomialmodell \cref{def:binModell} darstellt und somit die Binomialverteilung nach Definition eine Zähldichte definiert \cref{def:binZähldichte}. Daher gilt: $S_n \sim \mathcal{B}in_{n,p}$. + Nach \cref{bem:XiB} und \cref{def:Sn} ist $S_n$ die Summe von $n$ unabhängigen Bernoulli-verteilten Zufallsvariablen $X_i \sim \mathcal{B}_p$. Also stellt $S_n$ die Anzahl der Erfolge von n-Wiederholungen von unabhängigen identisch Bernoulli-verteilten Zufallsexperimenten dar. + Daraus folgt, dass das Galton-Brett mit der Zufallsvariable $S_n$ eine Binomialmodell \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} -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. -%Normalverteilung als Stetig Dichte einfphren +Eine weitere in diesem Kontext interessante Verteilung ist die Normalverteilung. \begin{definition}[Normalverteilung] - Inhalt... + Sei X eine stetige Zufallsvariable mit Dichte + \[ f(x)= \frac{1}{\sqrt{2\pi \sigma ^2}}\cdot e^{-\frac{\left( x-\mu \right)^2}{2\sigma ^2}}\] + 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: + \[ x \sim \mathcal{N}(\mu, \sigma ^2)\] \end{definition} -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} - 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 $S_n$ genau den Wert $k$ annimmt, 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 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: \[ - P(S_n = k) = \binom{n}{k} p^k q^{n-k} \approx \frac{1}{\sqrt{2\pi npq}} e^{\left(-\frac{(k-np)^2}{2npq}\right)} + P(S_n = k) \sim \mathcal{N}(\mu,\sigma ^2) \] + wobei $\mathbb{E}(X)=\mu$, $Var(X)=\sigma ^2$ 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.\\ @@ -92,9 +98,9 @@ Im Spezialfall der Binomialverteilung wird dies durch den Satz von Moivre-Laplac \[ P(S_n = k) = \frac{n!}{k!(n-k)!} p^k q^{n-k} \] - Nach \cref{satz:stirlingformel} gilt für große Zahlen näherungsweise $n! \approx \sqrt{2\pi n} \left(\frac{n}{e}\right)^n$. Ersetzen wir $n!, k!$ und $(n-k)!$ durch diese Näherung: + Nach \cref{satz:stirlingformel} gilt für große Zahlen näherungsweise $n! \sim \sqrt{2\pi n} \left(\frac{n}{e}\right)^n$. Ersetzen wir $n!, k!$ und $(n-k)!$ durch diese Näherung: \begin{align} - P(S_n = k) &\approx \frac{\sqrt{2\pi n} \left(\frac{n}{e}\right)^n}{(\sqrt{2\pi k} \left(\frac{k}{e}\right)^k)(\sqrt{2\pi (n-k)} \left(\frac{(n-k)}{e}\right)^{(n-k)})} p^k q^{n-k} \nonumber \\ + P(S_n = k) &\sim \frac{\sqrt{2\pi n} \left(\frac{n}{e}\right)^n}{(\sqrt{2\pi k} \left(\frac{k}{e}\right)^k)(\sqrt{2\pi (n-k)} \left(\frac{(n-k)}{e}\right)^{(n-k)})} p^k q^{n-k} \nonumber \\ &= \frac{n^n \sqrt{2\pi n}}{k^k \sqrt{2\pi k} (n-k)^{n-k} \sqrt{2\pi (n-k)}} p^k q^{n-k} \label{eq1:e}\\ &= \frac{\sqrt{n} \cdot n^k \cdot n^{n-k}}{\sqrt{2\pi} \sqrt{k} \sqrt{(n-k)} k^k (n-k)^{n-k}} p^k q^{n-k} \label{eq1:aufteilen}\\ &= \frac{1}{\sqrt{2\pi}}\sqrt{\frac{n}{k(n-k)}} \left(\frac{np}{k}\right)^k \left(\frac{nq}{n-k}\right)^{n-k} \label{eq1:zusammenfassen} @@ -106,6 +112,7 @@ Im Spezialfall der Binomialverteilung wird dies durch den Satz von Moivre-Laplac \item[(\ref{eq1:zusammenfassen})] Ist das Resultat nach dem Sortieren der Brüche anhand der Exponenten. \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 \[ Z_n=\frac{S_n-E(S_n)}{\sigma(S_n)} @@ -122,18 +129,18 @@ Im Spezialfall der Binomialverteilung wird dies durch den Satz von Moivre-Laplac \begin{align} k(n-k) &= (np + z\sqrt{npq})\cdot(nq - z\sqrt{npq}) \label{eq3:einsetzen}\\ &= n^2pq + (nz\sqrt{nqp}(q-p)-z^2npq) \label{eq3:umstellen}\\ - &\overset{n\rightarrow\infty}{=}n^2pq\label{eq3:absch} + &=n^2pq + O(n)\label{eq3:absch} \end{align} Im Folgenden werden die nummerierten Schritte der Gleichungsketten erläutert: \begin{itemize} \item[(\ref{eq3:einsetzen})] Die Formeln für $k$ und $n-k$ einsetzen. \item[(\ref{eq3:umstellen})] Das Distributivgesetz anwenden und nach der Potenz von $n$ sortieren. - \item[(\ref{eq3:absch})] $n^2$ geht für $n\rightarrow\infty$ schneller nach unendlich als die linearen Restterme. Folglich sind diese für die Abschätzung gegen unendlich vernachlässigbar. + \item[(\ref{eq3:absch})] $n^2$ geht für $n\rightarrow\infty$ schneller nach unendlich als die linearen Restterme. Folglich sind diese für die Abschätzung gegen unendlich im folgenden vernachlässigbar. \end{itemize} Setzen wir dieses Ergebnis nun in das Zwischenergebnis von \cref{eq1:zusammenfassen} ein, so erhalten wir folgendes: \begin{align} P(S_n = k) &= \frac{1}{\sqrt{2\pi}}\sqrt{\frac{n}{k(n-k)}} \left(\frac{np}{k}\right)^k \left(\frac{nq}{n-k}\right)^{n-k} \nonumber \\ - &\approx \frac{1}{\sqrt{2\pi}}\sqrt{\frac{n}{n^2pq}} \left(\frac{np}{k}\right)^k \left(\frac{nq}{n-k}\right)^{n-k} \label{eq4:1}\\ + &\sim \frac{1}{\sqrt{2\pi}}\sqrt{\frac{n}{n^2pq}} \left(\frac{np}{k}\right)^k \left(\frac{nq}{n-k}\right)^{n-k} \label{eq4:1}\\ &= \frac{1}{\sqrt{2\pi}}\frac{1}{\sqrt{npq}} \left(\frac{np}{k}\right)^k \left(\frac{nq}{n-k}\right)^{n-k} \label{eq4:2}\\ &= \frac{1}{\sqrt{2\pi npq}}\left(\frac{k}{np}\right)^{-k} \left(\frac{n-k}{nq}\right)^{-n+k} \label{eq4:3} \end{align} @@ -160,13 +167,13 @@ Im Spezialfall der Binomialverteilung wird dies durch den Satz von Moivre-Laplac \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} - Die \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. + Die \cref{def:taylorpolynom} beweist die Taylor-Approximation für den Logarithmus: $ln(1-\alpha) = -\alpha-\frac{\alpha^2}{2}+O(\alpha^3)$ und $ln(1+\alpha) \sim \alpha-\frac{\alpha^2}{2}+O(\alpha^3)$ für $\alpha \in (-\infty, 1)$. 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 $-1 <|\alpha| < 1$. 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.\\ + Um diese Approximation zu verwenden, muss sicher sein, dass $-1 < |\alpha| < 1$. Das heißt, dass $\forall z \in \mathbb{R}: \lim_{n\rightarrow\infty} -1 < |z\cdot\beta|<1$. (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\\ @@ -176,12 +183,12 @@ Im Spezialfall der Binomialverteilung wird dies durch den Satz von Moivre-Laplac Betrachten wir nun die Logarithmen für die Taylor-Approximation näher: \begin{align*} - \ln\left(1 + z\sqrt{\frac{q}{np}}\right) &\approx z\sqrt{\frac{q}{np}} - \frac{\left(z\sqrt{\frac{q}{np}}\right)^2}{2} + O((z\beta)^3)\\ + \ln\left(1 + z\sqrt{\frac{q}{np}}\right) &\sim z\sqrt{\frac{q}{np}} - \frac{\left(z\sqrt{\frac{q}{np}}\right)^2}{2} + O((z\beta)^3)\\ &= z\sqrt{\frac{q}{np}} - \frac{z^2 q}{2np} + O((z\beta)^3) \end{align*} und für den anderen ln analog: \begin{align*} - \ln\left(1 - z\sqrt{\frac{p}{nq}}\right) &\approx -z\sqrt{\frac{p}{nq}} - \frac{\left(z\sqrt{\frac{p}{nq}}\right)^2}{2} + O((-z\gamma)^3) \\ + \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: @@ -191,7 +198,7 @@ Im Spezialfall der Binomialverteilung wird dies durch den Satz von Moivre-Laplac &=(-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}\\ &=(-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} \\ - &\approx (-z\sqrt{npq}-\frac{z^2q}{2}) + (z\sqrt{npq}-\frac{z^2p}{2})\label{eq6:4}\\ + &\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}\\ &=-\frac{z^2}{2} \label{eq6:6} \end{align} %(-np - z\sqrt{npq}) \cdot O((z\beta)^3) + (-nq + z\sqrt{npq}) \cdot O((z\gamma)^3) @@ -200,39 +207,42 @@ Im Spezialfall der Binomialverteilung wird dies durch den Satz von Moivre-Laplac \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. Betrachte Hierzu \cref{lem:restterme} und \cref{prop:restterme2}. + \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{eq6:5})] $-z\sqrt{npq}+z\sqrt{npq}=0$ und Distributivgesetz (invers) anwenden. \item[(\ref{eq6:6})] Es gilt $p+q=1$, da $q = 1-p$ definiert wurde. \end{itemize} Da $e^{ln(l)} = l$ setzen wir nun $e^{-\frac{z^2}{2}}$ in \cref{eq4:3} ein, kommen wir zum finalen Ergebnis: \begin{align} - P(S_n = k) &\approx \frac{1}{\sqrt{2\pi npq}}e^{-\frac{z^2}{2}} \nonumber \\ + 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 npq}}e^{\left(-\frac{(k-np)^2}{2npq}\right)} \label{eq7:2} \\ + &= \mathcal{N}(np)(npq) = \mathcal{N}(\mu)(\sigma^2) \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. \end{itemize} Daher konvergiert eine 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 $z \in \mathbb{R}$ fest gewählt. 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 $z$ Werte einer standardisierten, binomialverteilten Zufallsvariable wobei z begrenzt ist. Für die Restterme der Taylor-Entwicklung des Logarithmus multipliziert mit ihren jeweiligen Vorfaktoren gilt für $n \rightarrow \infty$: \[ (-np - z\sqrt{npq}) \cdot O((z\beta)^3) + (-nq + z\sqrt{npq}) \cdot O((z\gamma)^3) \xrightarrow{n \rightarrow \infty} 0 \] wobei $\beta = \sqrt{\frac{q}{np}}$ und $\gamma = \sqrt{\frac{p}{nq}}$ definiert sind. \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. Multiplizieren wir dies mit dem Betrag des Vorfaktors, erhalten wir mithilfe der Dreiecksungleichung: \begin{align*} - \left| (-np - z\sqrt{npq}) \cdot O((z\beta)^3) \right| &\leq (np + |z|\sqrt{npq}) \cdot C \cdot n^{-3/2} \\ + \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} \end{align*} @@ -243,9 +253,9 @@ Im Spezialfall der Binomialverteilung wird dies durch den Satz von Moivre-Laplac 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 beliebig aber fest) 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 $z \in \mathbb{R}$ fest gewählt. 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 $z$ begrenzt ist. Für die Restterme der Taylor-Entwicklung des Logarithmus gilt für $n \rightarrow \infty$: \[ O((z\beta)^3) + O((z\gamma)^3) \xrightarrow{n \rightarrow \infty} 0 \] @@ -387,7 +397,7 @@ Nun können wir unsere Approximation von $\ln(1-x)=-x-\frac{x^2}{2}+O(x^3)$ zeig \begin{align*} \ln(1-x)&=-x-\frac{x^2}{2}+R_2(x) \end{align*} - wobei das Lagrange-Restglied etwa $R_2(x)\approx O(x^3)$, denn + wobei das Lagrange-Restglied etwa $R_2(x)\sim O(x^3)$, denn \begin{align*} R_2(x)&=\frac{1}{(2+1)!}\cdot(\ln(1-\xi))^{(2+1)}\cdot(x-0)^{(2+1)}\\ &=-\frac{1}{3(1-\xi)^3}\cdot x^3\quad\text{für ein }\xi\in(0,x)&&\text{\cref{lem:dreimalableiten}}\\ @@ -400,7 +410,7 @@ Nun können wir unsere Approximation von $\ln(1-x)=-x-\frac{x^2}{2}+O(x^3)$ zeig Da $\xi$ variabel ist (also von $x$ abhängt), steht $O(x^3)$ stellvertretend für ein gut passendes Polynom vom Grad 3. \[ - \Rightarrow\quad\ln(1-x)\approx -x -\frac{x^2}{2}+O(x^3) + \Rightarrow\quad\ln(1-x)\sim -x -\frac{x^2}{2}+O(x^3) \] \end{proof} \begin{bemerkung} @@ -435,7 +445,7 @@ Im Plot können wir sehen, dass beide Funktionen um unsere Entwicklungsstelle $x \end{tikzpicture} \caption{Darstellung der Funktionen $\ln(1-x)$ und $-x-\frac{x^2}{2}$} \end{figure} -Würden wir die Entwicklung weiterführen, approximiert die violette Funktion (unser Taylorpolynom) die orangene Funktion $(\ln(1-x))$ immer weiter. Graphisch lässt sich erkennen, dass für Werte im Intervall $I(-0.5,0.5)$ die Funktion $\ln(1-x) \approx -x\frac{-x}{2}$ plus ein kleiner Fehler ist. +Würden wir die Entwicklung weiterführen, approximiert die violette Funktion (unser Taylorpolynom) die orangene Funktion $(\ln(1-x))$ immer weiter. Graphisch lässt sich erkennen, dass für Werte im Intervall $I(-0.5,0.5)$ die Funktion $\ln(1-x) \sim -x\frac{-x}{2}$ plus ein kleiner Fehler ist. \subsection{Gamma-Funktion} % wird für Beweis der Stirling-Approximation gebraucht Zum nachfolgenden Betrachtung und des Beweises der Stirling-Approximation betrachten wir nun die Gamma-Funktion, auch \textbf{Eulersches Integral zweiter Gattung} genannt. Sie erweitert die Fakultätsfunktion von den natürlichen Zahlen $\mathbb{N}$ auf reelle und komplexe Zahlen (mit einigen Ausnahmen). Hier betrachten wir der Einfachheit halber nur die Gamma-Funktion in $\mathbb{R}$. @@ -460,7 +470,7 @@ Die Stirling-Approximation ist eine mathematische Näherung zur Berechnung der F \begin{satz}[Stirlingformel]{stirlingformel}{\cite{freitag1995-oj}} Für $n\mapsto\infty$ gilt nach \textsc{Stirlings} Approximation, dass \[ - n!\approx \sqrt{2\pi n}\left(\frac{n}{e}\right)^n + n!\sim \sqrt{2\pi n}\left(\frac{n}{e}\right)^n \] Das bedeutet \[ @@ -499,13 +509,13 @@ Die Stirling-Approximation ist eine mathematische Näherung zur Berechnung der F Damit besitzt \(f\) bei \(u=1\) einen stationären Punkt, und wir entwickeln \(f\) dort bis zur zweiten Ordnung: \[ - f(u)\approx f(1)+\frac{f''(1)}{2}(u-1)^2 + f(u)\sim f(1)+\frac{f''(1)}{2}(u-1)^2 = -1 - \frac{(u-1)^2}{2}. \] Einsetzen liefert die lokale Approximation \[ - \Gamma(x)\approx x^x e^{-x}\cdot\int_0^\infty \exp\!\left(-\frac{x}{2}(u-1)^2\right)\,du. + \Gamma(x)\sim x^x e^{-x}\cdot\int_0^\infty \exp\!\left(-\frac{x}{2}(u-1)^2\right)\,du. 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