add new slides

7 files changed, 319 insertions, 5 deletions

diff --git a/vorlesungen/slides/9/Makefile.inc b/vorlesungen/slides/9/Makefile.inc
index 4f8fc40..2ce78c0 100644
--- a/vorlesungen/slides/9/Makefile.inc
+++ b/vorlesungen/slides/9/Makefile.inc
@@ -18,5 +18,9 @@ chapter9 =								\
 	../slides/9/pf/vergleich3d.tex					\
 	../slides/9/pf/dreieck.tex					\
 	../slides/9/pf/folgerungen.tex					\
+	../slides/9/parrondo/erwartung.tex				\
+	../slides/9/parrondo/spiela.tex					\
+	../slides/9/parrondo/spielb.tex					\
+	../slides/9/parrondo/spielbmod.tex				\
 	../slides/9/chapter.tex
 
diff --git a/vorlesungen/slides/9/chapter.tex b/vorlesungen/slides/9/chapter.tex
index 1915073..86e528d 100644
--- a/vorlesungen/slides/9/chapter.tex
+++ b/vorlesungen/slides/9/chapter.tex
@@ -20,4 +20,8 @@
 \folie{9/pf/dreieck.tex}
 \folie{9/pf/folgerungen.tex}
 
+\folie{9/parrondo/erwartung.tex}
+\folie{9/parrondo/spiela.tex}
+\folie{9/parrondo/spielb.tex}
+\folie{9/parrondo/spielbmod.tex}
 
diff --git a/vorlesungen/slides/9/parrondo/erwartung.tex b/vorlesungen/slides/9/parrondo/erwartung.tex
new file mode 100644
index 0000000..67bb61d
--- /dev/null
+++ b/vorlesungen/slides/9/parrondo/erwartung.tex
@@ -0,0 +1,76 @@
+%
+% 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}
+\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}
+\begin{block}{Erwartungswerte}
+\begin{align*}
+E(X)
+&=
+\sum_i x_ip_i
+=
+x^tp
+=
+U^t x\odot p
+\\
+E(X^2)
+&=
+\sum_i x_i^2p_i
+=
+(x\odot x)^tp
+=
+U^t (x\odot x) \odot p
+\\
+E(X^k)
+&=
+\sum_i x_i^kp_i
+=
+U^t x^{\odot k}\odot p
+\end{align*}
+Substitution:
+\begin{align*}
+\sum_i &\to U^t\\
+x_i^k &\to x^{\odot k}
+\end{align*}
+Kann für Übergangsmatrizen von Markov-Ketten verallgemeinert werden
+\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..4b3b50c
--- /dev/null
+++ b/vorlesungen/slides/9/parrondo/spiela.tex
@@ -0,0 +1,51 @@
+%
+% 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+e
+\\
+P(X=         - 1)
+&=
+\frac12-e
+\end{align*}
+Bernoulli-Experiment mit $p=\frac12+e$
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Gewinnerwartung}
+\begin{align*}
+E(X)
+&=
+P(X=1)\cdot (1)
+\\
+&\qquad
++
+P(X=-1)\cdot (-1)
+\\
+&=
+\biggl(\frac12+e\biggr)\cdot 1
++
+\biggl(\frac12-e\biggr)\cdot (-1)
+\\
+&=2e
+\end{align*}
+$\Rightarrow$ {\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..6ad512c
--- /dev/null
+++ b/vorlesungen/slides/9/parrondo/spielb.tex
@@ -0,0 +1,83 @@
+%
+% 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}
+\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}
+\begin{block}{Markov-Kette $Y$}
+Übergangsmatrix
+\[
+B=\begin{pmatrix}
+0&\frac14&\frac34\\
+\frac{1}{10}&0&\frac14\\
+\frac{9}{10}&\frac34&0
+\end{pmatrix}
+\]
+Gewinnmatrix:
+\[
+G=\begin{pmatrix*}[r]
+0&-1&1\\
+1&0&-1\\
+-1&1&0
+\end{pmatrix*}
+\]
+\end{block}
+\begin{block}{Gewinnerwartung}
+\begin{align*}
+E(Y)
+&=
+U^t(G\odot B)p
+\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..ee1d12d
--- /dev/null
+++ b/vorlesungen/slides/9/parrondo/spielbmod.tex
@@ -0,0 +1,91 @@
+%
+% 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 $B$}
+\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 $\frac{9}{10}+\varepsilon$};
+\node at (150:{0.9*\r}) {\tiny $\frac1{10}-\varepsilon$};
+\node at (270:\r) {$\frac34-\varepsilon$};
+
+\node at (30:{1.1*\R}) {$\frac{3}{4}-\varepsilon$};
+\node at (150:{1.1*\R}) {$\frac1{4}+\varepsilon$};
+\node at (270:\R) {$\frac14+\varepsilon$};
+
+\end{tikzpicture}
+\end{center}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Markov-Kette $\tilde{Y}$}
+Übergangsmatrix
+\[
+\tilde{B}=
+B+\varepsilon F
+=
+B+\varepsilon\begin{pmatrix*}[r]
+0&1&-1\\
+-1&0&1\\
+1&-1&0
+\end{pmatrix*}
+\]
+Gewinnmatrix:
+\[
+G=\begin{pmatrix*}[r]
+0&-1&1\\
+1&0&-1\\
+-1&1&0
+\end{pmatrix*}
+\]
+\end{block}
+\begin{block}{Gewinnerwartung}
+\begin{align*}
+E(\tilde{Y})
+&=
+U^t(G\odot \tilde{B})p
+\\
+&=
+E(Y) + \varepsilon U^t(G\odot F)p
+=
+\frac1{15}+2\varepsilon
+\end{align*}
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/9/pf/positiv.tex b/vorlesungen/slides/9/pf/positiv.tex
index 382dfd6..d7e833d 100644
--- a/vorlesungen/slides/9/pf/positiv.tex
+++ b/vorlesungen/slides/9/pf/positiv.tex
@@ -18,9 +18,11 @@ a_{ij} > 0\quad\forall i,j
 \]
 Man schreibt $A>0\mathstrut$
 \end{block}
+\uncover<2->{%
 \begin{block}{Relation $>\mathstrut$}
 Man schreibt $A>B$ wenn $A-B > 0\mathstrut$
-\end{block}
+\end{block}}
+\uncover<5->{%
 \begin{block}{Wahrscheinlichkeitsmatrix}
 \[
 W=\begin{pmatrix}
@@ -30,19 +32,22 @@ W=\begin{pmatrix}
 \end{pmatrix}
 \]
 Spaltensumme$\mathstrut=1$, Zeilensumme$\mathstrut=?$
-\end{block}
+\end{block}}
 \end{column}
 \begin{column}{0.48\textwidth}
+\uncover<3->{%
 \begin{block}{Nichtnegative Matrix\strut}
 Eine Matrix $A$ heisst nichtnegativ, wenn
 \[
 a_{ij} \ge 0\quad\forall i,j
 \]
 Man schreibt $A\ge 0\mathstrut$
-\end{block}
+\end{block}}
+\uncover<4->{%
 \begin{block}{Relation $\ge\mathstrut$}
 Man schreibt $A\ge B$ wenn $A-B \ge 0\mathstrut$
-\end{block}
+\end{block}}
+\uncover<6->{%
 \begin{block}{Permutationsmatrix}
 \[
 P=\begin{pmatrix}
@@ -52,7 +57,7 @@ P=\begin{pmatrix}
 \end{pmatrix}
 \]
 Genau eine $1$ in jeder Zeile/Spalte
-\end{block}
+\end{block}}
 \end{column}
 \end{columns}
 \end{frame}