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diff --git a/vorlesungen/slides/dreieck/minmax.tex b/vorlesungen/slides/dreieck/minmax.tex new file mode 100644 index 0000000..9ef8d1a --- /dev/null +++ b/vorlesungen/slides/dreieck/minmax.tex @@ -0,0 +1,65 @@ +% +% minmax.tex -- Minimum und Maximum +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Minimum und Maximum} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Maximum} +Verteilungsfunktion von +\[ +Z=\operatorname{max}(X_1,\dots,X_n) +\] +\begin{align*} +F_Z(x) +&= +P(Z\le x) +\\ +&= +P(X_1\le x\wedge\dots\wedge X_n\le x) +\\ +&= +P(X_1\le x)\cdot \ldots\cdot P(X_n\le x) +\\ +&= +F_X(x)^n +\end{align*} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Minimum} +Verteilungsfunktion von +\[ +Z=\operatorname{min}(X_1,\dots,X_n) +\] +\begin{align*} +F_Z(x) +&= +P(Z\le x) +\\ +&=P(\overline{ +X_1\le x\wedge\dots\wedge X_n \le x +}) +\\ +&= +1-P( +X_1> x\wedge\dots\wedge X_n > x +) +\\ +&= +1-(P(X_1>x)\cdot\ldots\cdot P(X_n>x)) +\\ +&= +1-(1-F_X(x))^n +\end{align*} +\end{block} +\end{column} +\end{columns} +\end{frame} +\egroup |