Add most theory for exercise 2
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@ -31,9 +31,9 @@
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\usepackage{tikz-3dplot}
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\usetikzlibrary{spy, external, intersections, positioning}
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\ifdefined\ishandout\else
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\tikzexternalize
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\fi
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% \ifdefined\ishandout\else
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% \tikzexternalize
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% \fi
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\usepackage{pgfplots}
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\pgfplotsset{compat=newest}
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@ -666,44 +666,392 @@
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Theorie Wiederholung}
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\begin{frame}
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\frametitle{Gesetze großer Zahlen}
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\begin{itemize}
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\item Bernoulli'sches Gesetz großer Zahlen
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\item Chintschin'sches Gesetz großer Zahlen
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\end{itemize}
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% TODO: Write
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\end{frame}
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\begin{frame}
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\frametitle{Zentraler Grenzwertsatz (ZGWS)}
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\begin{itemize}
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\item ZGWS von Lindeberg-Lévy
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\item Grenzwertsatz von de Moivre-Laplace
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\end{itemize}
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\begin{itemize}
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\item Binomialverteilung Formeln
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\item ZGWS
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\item (?) ZGWS ist vom Konzept eine ``Erweiterung vom Gesetz
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großer Zahlen''
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\item (Binomialverteilung $\rightarrow$ Normalverteilung) Visualisierung
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\item Approximation der Binomialverteilung durch die
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Normalverteilung + wann zulässig
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\end{itemize}
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% TODO: Write
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\end{frame}
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\begin{frame}
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\frametitle{Erinnerung: Rechnen mit Normalverteilungen}
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% TODO: Write
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\end{frame}
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\begin{frame}
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\frametitle{Grenzwertsätze}
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\vspace*{-15mm}
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\begin{itemize}
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\item Chintschin'sches Gesetz großer Zahlen
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\end{itemize}%
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%
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% tex-fmt: off
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\begin{gather*}
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\left.
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\begin{array}{r}
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X_1, \ldots X_N \text{ unabhängig und identisch verteilt} \\
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E(X_1) < \infty
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\end{array}
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\right\}
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\hspace{5mm} \Rightarrow \hspace{5mm}
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\lim_{N \rightarrow \infty} P\mleft( \left\vert \frac{1}{N}
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\nsum_{n=1}^{N} X_N - E(X_1) \right\vert < \epsilon \mright) = 1
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\end{gather*}
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% tex-fmt: on
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\vspace*{5mm}
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\centering
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\begin{minipage}[t]{0.5\textwidth}
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\centering
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``Je mehr realisierungen betrachtet werden, desto
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wahrscheinlicher ist das arithmetische Mittel nah am
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Erwartungswert''
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\end{minipage}
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\vspace*{10mm}
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\pause
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\begin{itemize}
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\item Zentraler Grenzwertsatz von Lindeberg-Lévy
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\end{itemize}%
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%
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\begin{gather*}
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\left.
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\begin{array}{r}
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X_1, \ldots X_N \text{ unabhängig und identisch verteilt} \\
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E(X_1) < \infty \\
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V(X_1) < \infty
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\end{array}
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\right\}
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\hspace{5mm} \Rightarrow \hspace{5mm}
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\left\{
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\begin{array}{c}
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S_N = X_1 + \cdots + X_N, \hspace*{10mm} a < b \in
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\mathbb{R} \\[2mm]
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\displaystyle\lim_{N \rightarrow \infty} P\mleft( a \le \frac{S_N -
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N\mu}{\sqrt{N\sigma^2}} \le b \mright)
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= \Phi(b) - \Phi(a)
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\end{array}
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\right.
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\end{gather*}
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\vspace*{5mm}
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\centering
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\begin{minipage}[t]{0.7\textwidth}
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\centering
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``Die Summe unabhängiger und identisch verteilter ZV verhält
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sich immer mehr wie eine Normalverteilung, je mehr ZV
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betrachtet werden''
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\end{minipage}
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\end{frame}
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\begin{frame}
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\frametitle{Approximation einer Binomialverteilung\\ mit dem ZGWS}
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\vspace*{-5mm}
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\centering
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\begin{minipage}{0.5\textwidth}
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\begin{itemize}
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\item Grenzwertsatz von de Moivre-Laplace
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\begin{align*}
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\lim_{N \rightarrow \infty} P \mleft( \frac{S_N -
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Np}{\sqrt{Np(1-p)}} \le x \mright) = \Phi(x)
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\end{align*}
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\centering
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\vspace*{5mm}
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``$\text{Bin}(N,p) \overset{N \rightarrow
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\infty}{\relbar\joinrel\relbar\joinrel\relbar\joinrel\rightarrow}
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\mathcal{N}\big(\mu = Np, \sigma^2 = Np(1-p)\big)$''
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\end{itemize}
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\end{minipage}%
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\begin{minipage}{0.5\textwidth}
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\begin{lightgrayhighlightbox}
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\vspace*{-2mm}
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Errinerung: Binomialverteilung
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\begin{gather*}
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S_N \sim \text{\normalfont Bin}(N, p) \\[2mm]
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P_{S_N}(k) = \binom{N}{k} p^k (1-p)^{N-k} \\
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E(S_N) = Np, \hspace*{5mm} V(S_N) = Np(1-p)
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\end{gather*}
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\vspace*{-15mm}
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\end{lightgrayhighlightbox}
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\end{minipage}
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\pause
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\vspace{3mm}
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\begin{itemize}
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\item Die approximation einer Binomialverteilung durch eine
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Normalverteilung ist in der Praxis dann zulässig, \\
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wenn $Np(1-p) \ge 9$:
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\vspace{-2mm}
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\begin{align*}
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P_X(a < S_N \le b) = \nsum_{k=a}^{b} \binom{N}{k} p^k(1-p)^{N-k}
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\hspace{5mm}\approx\hspace{5mm}
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\Phi\left(\frac{b - Np}{\sqrt{Np(1-p)}}\right) -
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\Phi\left(\frac{a - Np}{\sqrt{Np(1-p)}}\right)
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\end{align*}
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\end{itemize}
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\pause
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\begin{minipage}[t]{0.32\textwidth}
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\centering
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\begin{figure}[H]
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\centering
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\begin{tikzpicture}
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\begin{axis}[
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width=10cm,
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height=5cm,
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scatter/classes={
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a={mark=*, blue}
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},
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xtick=\empty,
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ytick=\empty,
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xlabel = $x$,
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ylabel = $y$,
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xmin=-4,xmax=4,
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ymin=-1.5,ymax=2,
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]
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\addplot+[
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scol1,
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scatter,
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only marks,
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scatter src=explicit symbolic,
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]
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table[row sep=crcr] {
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x y \\
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0.9782846466992505 1.3425401677691273 \\
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-0.3342085827306991 -0.3478699656733771 \\
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1.0329768177464096 0.906099042791728 \\
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0.4032837175133078 0.09609805659133519 \\
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-0.47995152749835157 -0.5885801242458046 \\
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-0.39301528503877914 0.5165601264867574 \\
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-0.3016076234682761 0.3555224809310629 \\
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-1.283841439924361 -1.092505952596916 \\
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0.6394093134607625 0.7760543139022245 \\
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-1.3930746204117168 -1.2539179604346171 \\
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0.7066349809976303 0.26736104561273705 \\
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-0.32358511023766134 -0.4974120460927544 \\
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0.5697159086054595 0.7427982778218153 \\
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0.7810330322454977 1.021722205669364 \\
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-1.05027750818351 -1.088249765156553 \\
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-0.18753992607010203 -0.4808932985122092 \\
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0.9163016000620543 1.1130761981874584 \\
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-0.16588501836421943 -0.5254281720340348 \\
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1.7319708376031673 1.2174504869365954 \\
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-0.5732884092151935 -0.4923142548003758 \\
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1.2626814172655978 1.1468156532099922 \\
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-1.1007514357735002 -0.626200459957605 \\
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0.40631320003662 -0.3705703698506922 \\
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-2.221684738838144 -2.739364284431091 \\
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1.1309626619949467 1.1940429603335854 \\
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0.3055128861785891 0.529524240616076 \\
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-0.22522789028651527 -0.5082861632170081 \\
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0.2726524372676921 -0.2466404699684424 \\
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0.7078557266441373 0.8428284296154347 \\
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-1.4402649481540337 -0.9344326515164862 \\
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1.129522000340855 0.4510295424893529 \\
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0.5870764491195138 0.5669363454321612 \\
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2.3539677525351856 2.2253385575502285 \\
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0.2028654829519406 0.24539632425150296 \\
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-0.20861363707807395 -0.26125228812993867 \\
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0.6187802012217948 -0.2685299708916181 \\
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-0.2659232421672081 -0.22662166465228362 \\
|
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1.2403675794143405 1.0157380953006032 \\
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-0.0391562905128538 -0.6304153520459441 \\
|
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0.9833408241524402 1.06523679491654 \\
|
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0.5231710701516994 0.44339120385526315 \\
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-3.2527645820047146 -2.955881198077996 \\
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0.08993024102635327 0.6534559407213543 \\
|
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0.4076640826339743 0.5075313685387366 \\
|
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0.7431965606838403 0.04225691288802064 \\
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2.0420226454403996 2.411788877111026 \\
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-0.3652226483952774 -0.5846718876921133 \\
|
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0.478643720906727 0.7267990110235567 \\
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1.18457297115014 1.7548366300308906 \\
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0.05462743086401826 -0.02632310517996274\\
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};
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\end{axis}
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\end{tikzpicture}
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\end{figure}
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\end{minipage}%
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\begin{minipage}[t]{0.32\textwidth}
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\centering
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\begin{figure}[H]
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\centering
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\begin{tikzpicture}
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\begin{axis}[
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width=10cm,
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height=5cm,
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scatter/classes={
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a={mark=*, blue}
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},
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xtick=\empty,
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ytick=\empty,
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xlabel = $x$,
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ylabel = $y$,
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xmin=-2,xmax=1.7,
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ymin=-6,ymax=6,
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]
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\addplot+[
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scol1,
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scatter,
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only marks,
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scatter src=explicit symbolic,
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]
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table[row sep=crcr] {
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x y \\
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1.474648977967909 1.4746489779679088 \\
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0.16441809142886987 0.16441809142886984 \\
|
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-2.031822997202981 -2.0318229972029807 \\
|
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1.2182520939353458 1.2182520939353456 \\
|
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-0.5531479118291289 -0.5531479118291288 \\
|
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-1.47499216570319 -1.4749921657031897 \\
|
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0.6306758881059998 0.6306758881059997 \\
|
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-1.5640176651059605 -1.56401766510596 \\
|
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-1.3859450407248939 -1.3859450407248937 \\
|
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-1.0665944205353617 -1.0665944205353615 \\
|
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0.7492473817601838 0.7492473817601837 \\
|
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0.9499783172548493 0.949978317254849 \\
|
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0.34018738058932396 0.3401873805893239 \\
|
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-1.0096550061469416 -1.0096550061469414 \\
|
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-0.6754142249990085 -0.6754142249990084 \\
|
||||
-0.7514107790472649 -0.7514107790472647 \\
|
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0.7509769467458492 0.750976946745849 \\
|
||||
0.646133492401253 0.6461334924012528 \\
|
||||
-1.745989051249313 -1.7459890512493126 \\
|
||||
-0.29616542476720953 -0.2961654247672095 \\
|
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0.845676968386932 0.8456769683869317 \\
|
||||
-1.5863350761719144 -1.586335076171914 \\
|
||||
-0.42188175903423886 -0.42188175903423875 \\
|
||||
-1.6501793388980994 -1.650179338898099 \\
|
||||
-0.5605813083780707 -0.5605813083780706 \\
|
||||
0.9976042516699373 0.997604251669937 \\
|
||||
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|
||||
1.2028353996291026 1.2028353996291024 \\
|
||||
1.0123659351742527 1.0123659351742524 \\
|
||||
1.3398753611390357 1.3398753611390355 \\
|
||||
-1.8353880184343434 -1.835388018434343 \\
|
||||
0.20942595245430015 0.2094259524543001 \\
|
||||
-0.8821105204722243 -0.882110520472224 \\
|
||||
0.2823899445624717 0.28238994456247163 \\
|
||||
-0.6518146330435214 -0.6518146330435213 \\
|
||||
0.48917046669684383 0.4891704666968437 \\
|
||||
0.8356612510810297 0.8356612510810295 \\
|
||||
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|
||||
0.2947454674362563 0.29474546743625624 \\
|
||||
-1.2159944359594155 -1.2159944359594153 \\
|
||||
-0.21681449173721903 -0.21681449173721898 \\
|
||||
-0.7152726760226845 -0.7152726760226844 \\
|
||||
1.3162334939362095 1.3162334939362093 \\
|
||||
0.2527835241858269 0.25278352418582684 \\
|
||||
-0.9986661858536788 -0.9986661858536786 \\
|
||||
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|
||||
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|
||||
-2.30328755152605 -2.3032875515260494 \\
|
||||
-1.2929155901531633 -1.292915590153163 \\
|
||||
-0.8530475462391427 -0.8530475462391425\\
|
||||
};
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||||
\end{axis}
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||||
\end{tikzpicture}
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\end{figure}
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\end{minipage}%
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\begin{minipage}[t]{0.32\textwidth}
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\centering
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\begin{figure}[H]
|
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\centering
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|
||||
\begin{tikzpicture}
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||||
\begin{axis}[
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width=10cm,
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||||
height=5cm,
|
||||
scatter/classes={
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a={mark=*, blue}
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||||
},
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||||
xtick=\empty,
|
||||
ytick=\empty,
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||||
xlabel = $x$,
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||||
ylabel = $y$,
|
||||
xmin=-4,xmax=4,
|
||||
ymin=-2,ymax=2,
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]
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||||
\addplot+[
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scol1,
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scatter,
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only marks,
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scatter src=explicit symbolic,
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]
|
||||
table[row sep=crcr] {
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x y \\
|
||||
-1.7077500965534018 0.9715072286946655 \\
|
||||
1.1148806736392152 0.9230117606631614 \\
|
||||
0.11058932943085453 -0.5157596318522968 \\
|
||||
0.08527262614233909 -0.9720863462538933 \\
|
||||
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|
||||
-0.6705061079694784 -0.061860055599544606 \\
|
||||
-0.6212830814536863 1.2589504540208847 \\
|
||||
1.4236240086652356 -1.302789472184279 \\
|
||||
-1.0975355477486632 -0.886909899437918 \\
|
||||
1.5752231220795536 1.2260114881873 \\
|
||||
1.2049717160174165 1.0705757620706944 \\
|
||||
-1.7929521084203113 -1.0124364432205855 \\
|
||||
1.1345482934601252 -0.7213210134187505 \\
|
||||
0.06993810174580865 -1.5278087661910722 \\
|
||||
0.50560442840041 1.1191719084519776 \\
|
||||
-0.814167507403749 0.2021470144855546 \\
|
||||
2.03061011925002 0.08990067866176893 \\
|
||||
0.7257818062658367 0.22602273591014058 \\
|
||||
0.5036942935085902 0.2520250465804246 \\
|
||||
0.5973644458579076 -0.2093760967114109 \\
|
||||
1.1104283164930224 1.5071527221448955 \\
|
||||
-0.052216510646198096 -0.5465573566030532 \\
|
||||
0.423205976943666 -0.21077815853809784 \\
|
||||
0.2982451040844636 -1.3591258564459687 \\
|
||||
0.539438662504297 -0.780387830281188 \\
|
||||
0.08417174139937453 0.2725275842632153 \\
|
||||
0.05733773656028022 0.8226842222044897 \\
|
||||
0.12184004421107687 -1.0962860273484687 \\
|
||||
3.0973129011059326 0.13325075656192403 \\
|
||||
1.464718817591499 -2.0541680373660234 \\
|
||||
0.6017327837974983 -0.43330515099025413 \\
|
||||
-1.6527036180073127 -0.04153499563379528 \\
|
||||
1.3583641617521591 -0.9127837751641491 \\
|
||||
-0.2808122864213532 0.6566355071818034 \\
|
||||
0.36085503878766245 -0.2372816111687184 \\
|
||||
-0.7808961491915221 -0.4569496546349541 \\
|
||||
-0.08144830754364803 0.5297194167082963 \\
|
||||
-0.3832453043478111 0.695158762430314 \\
|
||||
-0.3021005547959829 -0.7515146005101381 \\
|
||||
0.0832540012145203 -1.6257847886861803 \\
|
||||
-0.08783078629061673 0.48401963778829576 \\
|
||||
0.5098330610876248 0.3327688893197499 \\
|
||||
0.4804292632122983 -0.5397408326625166 \\
|
||||
0.3612454424603153 -0.2728088913965057 \\
|
||||
0.8706855868841972 1.8337909595106936 \\
|
||||
-0.7868151662161218 1.643221471861054 \\
|
||||
-0.5629480754112661 0.16190044666568626 \\
|
||||
0.9623486507086952 -0.06821392925238735 \\
|
||||
-0.390445497949156 -1.4902360360777431 \\
|
||||
2.239228377147278 -0.2037307482272916\\
|
||||
};
|
||||
\end{axis}
|
||||
\end{tikzpicture}
|
||||
\end{figure}
|
||||
\end{minipage}%
|
||||
|
||||
\vspace*{3mm}
|
||||
\begin{minipage}[t]{0.32\textwidth}
|
||||
\centering
|
||||
$N = 10$
|
||||
\end{minipage}%
|
||||
\begin{minipage}[t]{0.32\textwidth}
|
||||
\centering
|
||||
$N = 100$
|
||||
\end{minipage}%
|
||||
\begin{minipage}[t]{0.32\textwidth}
|
||||
\centering
|
||||
$N = 1000$
|
||||
\end{minipage}%
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}
|
||||
\frametitle{Zusammenfassung}
|
||||
|
||||
|
||||
Loading…
Reference in New Issue
Block a user