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+%
+% 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