From 0716f027660d55f30442bf2d8ede2ac3a0a4f175 Mon Sep 17 00:00:00 2001 From: Andreas Tsouchlos Date: Wed, 21 Jan 2026 17:51:32 +0100 Subject: [PATCH] Add most theory for exercise 2 --- src/2026-01-30/presentation.tex | 418 +++++++++++++++++++++++++++++--- 1 file changed, 383 insertions(+), 35 deletions(-) diff --git a/src/2026-01-30/presentation.tex b/src/2026-01-30/presentation.tex index db68e52..b80d268 100644 --- a/src/2026-01-30/presentation.tex +++ b/src/2026-01-30/presentation.tex @@ -31,9 +31,9 @@ \usepackage{tikz-3dplot} \usetikzlibrary{spy, external, intersections, positioning} -\ifdefined\ishandout\else -\tikzexternalize -\fi +% \ifdefined\ishandout\else +% \tikzexternalize +% \fi \usepackage{pgfplots} \pgfplotsset{compat=newest} @@ -666,44 +666,392 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Theorie Wiederholung} -\begin{frame} - \frametitle{Gesetze großer Zahlen} - - \begin{itemize} - \item Bernoulli'sches Gesetz großer Zahlen - \item Chintschin'sches Gesetz großer Zahlen - \end{itemize} - - % TODO: Write -\end{frame} - -\begin{frame} - \frametitle{Zentraler Grenzwertsatz (ZGWS)} - - \begin{itemize} - \item ZGWS von Lindeberg-Lévy - \item Grenzwertsatz von de Moivre-Laplace - \end{itemize} - - \begin{itemize} - \item Binomialverteilung Formeln - \item ZGWS - \item (?) ZGWS ist vom Konzept eine ``Erweiterung vom Gesetz - großer Zahlen'' - \item (Binomialverteilung $\rightarrow$ Normalverteilung) Visualisierung - \item Approximation der Binomialverteilung durch die - Normalverteilung + wann zulässig - \end{itemize} - - % TODO: Write -\end{frame} - \begin{frame} \frametitle{Erinnerung: Rechnen mit Normalverteilungen} % TODO: Write \end{frame} +\begin{frame} + \frametitle{Grenzwertsätze} + + \vspace*{-15mm} + + \begin{itemize} + \item Chintschin'sches Gesetz großer Zahlen + \end{itemize}% + % + % tex-fmt: off + \begin{gather*} + \left. + \begin{array}{r} + X_1, \ldots X_N \text{ unabhängig und identisch verteilt} \\ + E(X_1) < \infty + \end{array} + \right\} + \hspace{5mm} \Rightarrow \hspace{5mm} + \lim_{N \rightarrow \infty} P\mleft( \left\vert \frac{1}{N} + \nsum_{n=1}^{N} X_N - E(X_1) \right\vert < \epsilon \mright) = 1 + \end{gather*} + % tex-fmt: on + + \vspace*{5mm} + \centering + \begin{minipage}[t]{0.5\textwidth} + \centering + ``Je mehr realisierungen betrachtet werden, desto + wahrscheinlicher ist das arithmetische Mittel nah am + Erwartungswert'' + \end{minipage} + + \vspace*{10mm} + \pause + \begin{itemize} + \item Zentraler Grenzwertsatz von Lindeberg-Lévy + \end{itemize}% + % + \begin{gather*} + \left. + \begin{array}{r} + X_1, \ldots X_N \text{ unabhängig und identisch verteilt} \\ + E(X_1) < \infty \\ + V(X_1) < \infty + \end{array} + \right\} + \hspace{5mm} \Rightarrow \hspace{5mm} + \left\{ + \begin{array}{c} + S_N = X_1 + \cdots + X_N, \hspace*{10mm} a < b \in + \mathbb{R} \\[2mm] + \displaystyle\lim_{N \rightarrow \infty} P\mleft( a \le \frac{S_N - + N\mu}{\sqrt{N\sigma^2}} \le b \mright) + = \Phi(b) - \Phi(a) + \end{array} + \right. + \end{gather*} + + \vspace*{5mm} + \centering + \begin{minipage}[t]{0.7\textwidth} + \centering + ``Die Summe unabhängiger und identisch verteilter ZV verhält + sich immer mehr wie eine Normalverteilung, je mehr ZV + betrachtet werden'' + \end{minipage} +\end{frame} + +\begin{frame} + \frametitle{Approximation einer Binomialverteilung\\ mit dem ZGWS} + + \vspace*{-5mm} + + \centering + \begin{minipage}{0.5\textwidth} + \begin{itemize} + \item Grenzwertsatz von de Moivre-Laplace + \begin{align*} + \lim_{N \rightarrow \infty} P \mleft( \frac{S_N - + Np}{\sqrt{Np(1-p)}} \le x \mright) = \Phi(x) + \end{align*} + + \centering + \vspace*{5mm} + ``$\text{Bin}(N,p) \overset{N \rightarrow + \infty}{\relbar\joinrel\relbar\joinrel\relbar\joinrel\rightarrow} + \mathcal{N}\big(\mu = Np, \sigma^2 = Np(1-p)\big)$'' + \end{itemize} + \end{minipage}% + \begin{minipage}{0.5\textwidth} + \begin{lightgrayhighlightbox} + \vspace*{-2mm} + Errinerung: Binomialverteilung + \begin{gather*} + S_N \sim \text{\normalfont Bin}(N, p) \\[2mm] + P_{S_N}(k) = \binom{N}{k} p^k (1-p)^{N-k} \\ + E(S_N) = Np, \hspace*{5mm} V(S_N) = Np(1-p) + \end{gather*} + \vspace*{-15mm} + \end{lightgrayhighlightbox} + \end{minipage} + + \pause + \vspace{3mm} + \begin{itemize} + \item Die approximation einer Binomialverteilung durch eine + Normalverteilung ist in der Praxis dann zulässig, \\ + wenn $Np(1-p) \ge 9$: + \vspace{-2mm} + \begin{align*} + P_X(a < S_N \le b) = \nsum_{k=a}^{b} \binom{N}{k} p^k(1-p)^{N-k} + \hspace{5mm}\approx\hspace{5mm} + \Phi\left(\frac{b - Np}{\sqrt{Np(1-p)}}\right) - + \Phi\left(\frac{a - Np}{\sqrt{Np(1-p)}}\right) + \end{align*} + \end{itemize} + + \pause + \begin{minipage}[t]{0.32\textwidth} + \centering + \begin{figure}[H] + \centering + + \begin{tikzpicture} + \begin{axis}[ + width=10cm, + height=5cm, + scatter/classes={ + a={mark=*, blue} + }, + xtick=\empty, + ytick=\empty, + xlabel = $x$, + ylabel = $y$, + xmin=-4,xmax=4, + ymin=-1.5,ymax=2, + ] + \addplot+[ + scol1, + scatter, + only marks, + scatter src=explicit symbolic, + ] + table[row sep=crcr] { + x y \\ + 0.9782846466992505 1.3425401677691273 \\ + -0.3342085827306991 -0.3478699656733771 \\ + 1.0329768177464096 0.906099042791728 \\ + 0.4032837175133078 0.09609805659133519 \\ + 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