%
% dichte.tex -- Wahrscheinlichkeitsdichte
%
% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
%
\bgroup
\definecolor{darkgreen}{rgb}{0,0.6,0}
\begin{frame}[t]
\setlength{\abovedisplayskip}{5pt}
\setlength{\belowdisplayskip}{5pt}
\frametitle{Wahrscheinlichkeitsdichte}
\vspace{-20pt}
\begin{columns}[t,onlytextwidth]
\begin{column}{0.40\textwidth}
\begin{block}{Definition}
\[
\varphi_{X_{k:n}}(x)
=
\frac{d}{dx} F_{X_{k:n}}(x)
\]
\end{block}
\end{column}
\begin{column}{0.60\textwidth}
\uncover<4->{%
\begin{block}{Gleichverteilung}
\[
{\color{darkgreen}F(x)}=\begin{cases}
0&x \le 0\\
x&0\le x \le 1,\\
1&x\ge 1
\end{cases}
\quad
\uncover<5->{
{\color{red}\varphi(x)}
=
\begin{cases}
1&0\le x \le 1\\
0&\text{sonst}
\end{cases}}
\]
\end{block}}
\end{column}
\end{columns}
\uncover<2->{%
\begin{block}{Ordnungsstatistik}
nach einiger Rechnung:
\begin{align*}
\varphi_{X_{k:n}}(x)
&=
{\color<3->{red}\varphi_X(x)}\,k\binom{n}{k}{\color<3->{darkgreen}F_X(x)}^{k-1}
(1-{\color<3->{darkgreen}F_X(x)})^{n-k}
\intertext{\uncover<4->{für Gleichverteilung}}
\uncover<6->{
\varphi_{X_{k:n}}(x)
&=
\begin{cases}
\displaystyle
{\color<7->{blue}k\binom{n}{k}}{\color{darkgreen}x}^{k-1}(1-{\color{darkgreen}x})^{n-k}
&0\le x \le 1\\
0&\text{sonst}
\end{cases}
\qquad\uncover<7->{\text{({\color{blue}Normierung})}}
}
\end{align*}
\end{block}}
\end{frame}
\egroup