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author | Andreas Müller <andreas.mueller@ost.ch> | 2021-05-20 10:09:26 +0200 |
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committer | Andreas Müller <andreas.mueller@ost.ch> | 2021-05-20 10:09:26 +0200 |
commit | 80416f0ab893f2b80a01be4acc13bd03c7a03682 (patch) | |
tree | 2f81661e07022daa69dbedbda14d3ce827b6b52d /vorlesungen/slides/9/parrondo/erwartung.tex | |
parent | fix handout (diff) | |
download | SeminarMatrizen-80416f0ab893f2b80a01be4acc13bd03c7a03682.tar.gz SeminarMatrizen-80416f0ab893f2b80a01be4acc13bd03c7a03682.zip |
add new slides
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-rw-r--r-- | vorlesungen/slides/9/parrondo/erwartung.tex | 76 |
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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 |