diff options
Diffstat (limited to 'vorlesungen/slides/9/parrondo')
| -rw-r--r-- | vorlesungen/slides/9/parrondo/erwartung.tex | 76 | ||||
| -rw-r--r-- | vorlesungen/slides/9/parrondo/spiela.tex | 51 | ||||
| -rw-r--r-- | vorlesungen/slides/9/parrondo/spielb.tex | 83 | ||||
| -rw-r--r-- | vorlesungen/slides/9/parrondo/spielbmod.tex | 91 |
4 files changed, 301 insertions, 0 deletions
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 |