Merge branch 'master' into master

7 files changed, 471 insertions, 0 deletions

diff --git a/vorlesungen/slides/9/parrondo/deformation.tex b/vorlesungen/slides/9/parrondo/deformation.tex
new file mode 100644
index 0000000..40d2eb9
--- /dev/null
+++ b/vorlesungen/slides/9/parrondo/deformation.tex
@@ -0,0 +1,45 @@
+%
+% deformation.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Deformation}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Verlustspiele}
+Durch Deformation (Parameter $e$ und $\varepsilon$) kann man
+aus $A_e$ und $B_\varepsilon$ Spiele mit negativer Gewinnerwartung machen
+\uncover<2->{%
+\begin{align*}
+E(X)&=0&&\rightarrow&E(X_e)&<0\\
+E(Y)&=0&&\rightarrow&E(Y_\varepsilon)&<0\\
+\end{align*}}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Kombiniertes Spiel}
+\uncover<3->{%
+Die Deformation für das Spiel $C$ startet mit Erwartungswert $\frac{18}{709}$}%
+\begin{align*}
+\uncover<4->{E(Z)&=\frac{18}{709}>0}
+&&\uncover<5->{\rightarrow&
+E(Z_*)&>0}
+\end{align*}
+\uncover<6->{Wegen Stetigkeit!}
+\\
+\uncover<5->{Die Deformation ist immer noch ein Gewinnspiel (für Parameter klein genug)}
+\end{block}
+\uncover<7->{%
+\begin{block}{Parrondo-Paradoxon}
+Zufällig zwischen zwei Verlustspielen auswählen kann trotzdem ein
+Gewinnspiel ergeben
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/9/parrondo/erwartung.tex b/vorlesungen/slides/9/parrondo/erwartung.tex
new file mode 100644
index 0000000..b58c37f
--- /dev/null
+++ b/vorlesungen/slides/9/parrondo/erwartung.tex
@@ -0,0 +1,81 @@
+%
+% erwartung.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Erwartung}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Zufallsvariable}
+\begin{center}
+\[
+\begin{array}{c|c}
+\text{Werte $X$}&\text{Wahrscheinlichkeit $p$}\\
+\hline
+x_1&p_1=P(X=x_1)\\
+x_2&p_2=P(X=x_2)\\
+\vdots&\vdots\\
+x_n&p_n=P(X=x_n)
+\end{array}
+\]
+\end{center}
+\end{block}
+\uncover<4->{%
+\begin{block}{Einervektoren/-matrizen}
+\[
+U=\begin{pmatrix}
+1&1&\dots&1\\
+1&1&\dots&1\\
+\vdots&\vdots&\ddots&\vdots\\
+1&1&\dots&1
+\end{pmatrix}
+\in
+M_{n\times m}(\Bbbk)
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<2->{%
+\begin{block}{Erwartungswerte}
+\begin{align*}
+E(X)
+&=
+\sum_i x_ip_i
+=
+x^tp
+\uncover<5->{=
+U^t x\odot p}
+\hspace*{3cm}
+\\
+\uncover<2->{E(X^2)
+&=
+\sum_i x_i^2p_i}
+\ifthenelse{\boolean{presentation}}{
+\only<6>{=
+(x\odot x)^tp}}{}
+\uncover<7->{=
+U^t (x\odot x) \odot p}
+\\
+\uncover<3->{E(X^k)
+&=
+\sum_i x_i^kp_i}
+\uncover<8->{=
+U^t x^{\odot k}\odot p}
+\end{align*}
+\uncover<9->{%
+Substitution:
+\begin{align*}
+\uncover<10->{\sum_i &\to U^t}\\
+\uncover<11->{x_i^k &\to x^{\odot k}}
+\end{align*}}%
+\uncover<12->{Kann für Übergangsmatrizen von Markov-Ketten verallgemeinert werden}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/9/parrondo/kombiniert.tex b/vorlesungen/slides/9/parrondo/kombiniert.tex
new file mode 100644
index 0000000..5012d06
--- /dev/null
+++ b/vorlesungen/slides/9/parrondo/kombiniert.tex
@@ -0,0 +1,73 @@
+%
+% kombiniert.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Kombiniertes Spiel $C$}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Definition}
+Ein fairer Münzwurf entscheidet, ob
+Spiel $A$ oder Spiel $B$ gespielt wird
+\end{block}
+\uncover<2->{%
+\begin{block}{Übergangsmatrix}
+Münzwurf $X$
+\begin{align*}
+C
+&=
+P(X=\text{Kopf})\cdot A
++
+P(X=\text{Zahl})\cdot B
+\\
+&\uncover<3->{=
+\begin{pmatrix}
+           0&\frac{3}{8}&\frac{5}{8}\\
+\frac{3}{10}&          0&\frac{3}{8}\\
+\frac{7}{10}&\frac{5}{8}&          0
+\end{pmatrix}}
+\end{align*}
+\end{block}}
+\vspace{-8pt}
+\uncover<4->{%
+\begin{block}{Gewinnerwartung im Einzelspiel}
+\[
+p=\frac13U
+\Rightarrow
+U^t(G\odot C)p
+\uncover<5->{=
+-\frac{1}{30}}
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<6->{%
+\begin{block}{Iteriertes Spiel}
+\[
+\overline{p}=C\overline{p}
+\quad
+\uncover<7->{\Rightarrow
+\quad
+\overline{p}=\frac{1}{709}\begin{pmatrix}245\\180\\284\end{pmatrix}}
+\]
+\end{block}}
+\uncover<8->{%
+\begin{block}{Gewinnerwartung}
+\begin{align*}
+E(Z)
+&=
+U^t (G\odot C) \overline{p}
+\uncover<9->{=
+\frac{18}{709}}
+\end{align*}
+\uncover<10->{$C$ ist ein Gewinnspiel!}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/9/parrondo/spiela.tex b/vorlesungen/slides/9/parrondo/spiela.tex
new file mode 100644
index 0000000..629586f
--- /dev/null
+++ b/vorlesungen/slides/9/parrondo/spiela.tex
@@ -0,0 +1,52 @@
+%
+% spiela.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Spiel $A$}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Definition}
+Gewinn = Zufallsvariable $X$ mit Werten $\pm 1$
+\begin{align*}
+P(X=\phantom{+}1)
+&=
+\frac12\uncover<2->{+e}
+\\
+P(X=         - 1)
+&=
+\frac12\uncover<2->{-e}
+\end{align*}
+Bernoulli-Experiment mit $p=\frac12\uncover<2->{+e}$
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<3->{
+\begin{block}{Gewinnerwartung}
+\begin{align*}
+E(X)
+&=\uncover<4->{
+P(X=1)\cdot (1)}
+\\
+&\qquad
+\uncover<4->{+
+P(X=-1)\cdot (-1)}
+\\
+&\uncover<5->{=
+\biggl(\frac12+e\biggr)\cdot 1
++
+\biggl(\frac12-e\biggr)\cdot (-1)}
+\\
+&\uncover<6->{=2e}
+\end{align*}
+\uncover<7->{$\Rightarrow$ {\usebeamercolor[fg]{title}Verlustspiel für $e<0$}}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/9/parrondo/spielb.tex b/vorlesungen/slides/9/parrondo/spielb.tex
new file mode 100644
index 0000000..f65564f
--- /dev/null
+++ b/vorlesungen/slides/9/parrondo/spielb.tex
@@ -0,0 +1,100 @@
+%
+% spielb.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Spiel $B$}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Definition}
+Gewinn $\pm 1$, Wahrscheinlichkeit abhängig vom 3er-Rest des
+aktuellen Kapitals $K$:
+\begin{center}
+\uncover<2->{%
+\begin{tikzpicture}[>=latex,thick]
+\coordinate (A0) at (90:2);
+\coordinate (A1) at (210:2);
+\coordinate (A2) at (330:2);
+
+\node at (A0) {$0$};
+\node at (A1) {$1$};
+\node at (A2) {$2$};
+
+\draw (A0) circle[radius=0.4];
+\draw (A1) circle[radius=0.4];
+\draw (A2) circle[radius=0.4];
+
+\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A0) -- (A1);
+\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A0) -- (A2);
+\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A1) -- (A2);
+
+\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A1) to[out=90,in=-150] (A0);
+\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A2) to[out=90,in=-30] (A0);
+\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A2) to[out=-150,in=-30] (A1);
+
+\def\R{1.9}
+\def\r{0.7}
+
+\node at (30:\r) {$\frac{9}{10}$};
+\node at (150:\r) {$\frac1{10}$};
+\node at (270:\r) {$\frac34$};
+
+\node at (30:\R) {$\frac{3}{4}$};
+\node at (150:\R) {$\frac1{4}$};
+\node at (270:\R) {$\frac14$};
+
+\end{tikzpicture}}
+\end{center}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<3->{%
+\begin{block}{Markov-Kette $Y$}
+Übergangsmatrix
+\[
+B=\begin{pmatrix}
+0&\frac14&\frac34\\
+\frac{1}{10}&0&\frac14\\
+\frac{9}{10}&\frac34&0
+\end{pmatrix}
+\]
+\vspace{-10pt}
+
+\uncover<4->{%
+Gewinnmatrix:
+\vspace{-2pt}
+\[
+G=\begin{pmatrix*}[r]
+0&-1&1\\
+1&0&-1\\
+-1&1&0
+\end{pmatrix*}
+\]}
+\end{block}}
+\vspace{-12pt}
+\uncover<5->{%
+\begin{block}{Gewinnerwartung}
+\begin{align*}
+&&&&
+E(Y)
+&=
+U^t(G\odot B)p
+\\
+p&={\textstyle\frac13}U
+&&\Rightarrow&
+E(Y)&={\textstyle\frac1{15}}
+\\
+\overline{p}&={\tiny\frac{1}{13}\begin{pmatrix}5\\2\\6\end{pmatrix}}
+&&\Rightarrow&
+E(Y)&=0
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/9/parrondo/spielbmod.tex b/vorlesungen/slides/9/parrondo/spielbmod.tex
new file mode 100644
index 0000000..66d39bc
--- /dev/null
+++ b/vorlesungen/slides/9/parrondo/spielbmod.tex
@@ -0,0 +1,103 @@
+%
+% spielb.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Modifiziertes Spiel $\tilde{B}$}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Definition}
+Gewinn $\pm 1$, Wahrscheinlichkeit abhängig vom 3er-Rest des
+aktuellen Kapitals $K$:
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\coordinate (A0) at (90:2);
+\coordinate (A1) at (210:2);
+\coordinate (A2) at (330:2);
+
+\node at (A0) {$0$};
+\node at (A1) {$1$};
+\node at (A2) {$2$};
+
+\draw (A0) circle[radius=0.4];
+\draw (A1) circle[radius=0.4];
+\draw (A2) circle[radius=0.4];
+
+\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A0) -- (A1);
+\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A0) -- (A2);
+\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A1) -- (A2);
+
+\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A1) to[out=90,in=-150] (A0);
+\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A2) to[out=90,in=-30] (A0);
+\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A2) to[out=-150,in=-30] (A1);
+
+\def\R{1.9}
+\def\r{0.7}
+
+\node at (30:{0.9*\r}) {\tiny $\frac{9}{10}\uncover<2->{+\varepsilon}$};
+\node at (150:{0.9*\r}) {\tiny $\frac1{10}\uncover<2->{-\varepsilon}$};
+\node at (270:\r) {$\frac34\uncover<2->{-\varepsilon}$};
+
+\node at (30:{1.1*\R}) {$\frac{3}{4}\uncover<2->{-\varepsilon}$};
+\node at (150:{1.1*\R}) {$\frac1{4}\uncover<2->{+\varepsilon}$};
+\node at (270:\R) {$\frac14\uncover<2->{+\varepsilon}$};
+
+\end{tikzpicture}
+\end{center}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Markov-Kette $\tilde{Y}$}
+Übergangsmatrix
+\[
+\tilde{B}=
+B\uncover<2->{+\varepsilon F}
+\uncover<3->{=
+B+\varepsilon\begin{pmatrix*}[r]
+0&1&-1\\
+-1&0&1\\
+1&-1&0
+\end{pmatrix*}}
+\]
+\vspace{-12pt}
+
+\uncover<4->{%
+Gewinnmatrix:
+\[
+G=\begin{pmatrix*}[r]
+0&-1&1\\
+1&0&-1\\
+-1&1&0
+\end{pmatrix*}
+\]}
+\end{block}
+\vspace{-12pt}
+\uncover<5->{%
+\begin{block}{Gewinnerwartung}
+\begin{align*}
+\uncover<6->{E(\tilde{Y})
+&=
+U^t(G\odot \tilde{B})p}
+\\
+&\uncover<7->{=
+E(Y) + \varepsilon U^t(G\odot F)p}
+\uncover<8->{=
+{\textstyle\frac1{15}}+2\varepsilon}
+\\
+\uncover<9->{
+\text{rep.}
+&=
+-{\textstyle\frac{294}{169}}\varepsilon+O(\varepsilon^2)
+\quad\text{Verlustspiel}
+}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/9/parrondo/uebersicht.tex b/vorlesungen/slides/9/parrondo/uebersicht.tex
new file mode 100644
index 0000000..2f3597a
--- /dev/null
+++ b/vorlesungen/slides/9/parrondo/uebersicht.tex
@@ -0,0 +1,17 @@
+%
+% uebersicht.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Parrondo-Paradoxon}
+\begin{center}
+\Large
+Zufällige
+Wahl zwischen zwei Verlustspielen = Gewinnspiel?
+\end{center}
+\end{frame}
+\egroup