Finish explanation slides
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@ -46,6 +46,7 @@
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\usepackage{amsmath}
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\usepackage{graphicx}
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\usepackage{calc}
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\usepackage{amssymb}
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\title{WT Tutorium 5}
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\author[Tsouchlos]{Andreas Tsouchlos}
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@ -98,18 +99,17 @@
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\subsection{Theorie Wiederholung}
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\begin{frame}
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\frametitle{Unabhängige Zufallsvariablen}
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\frametitle{Summen Unabhängiger Zufallsvariablen}
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\begin{itemize}
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\item Korrelation $\ne$ Unabhängigkeit (außer bei Normalverteilung)
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\item Faltungssatz
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\item Charakteristische Funktion für Summen
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\end{itemize}
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\begin{itemize}
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\item Unabhängigkeit hat nichts mit den Einzelverteilungen zu
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tun, sie ist ``eine Ebene höher''
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\end{itemize}
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\begin{gather*}
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Z = X + Y, \hspace{10mm}X,Y \text{ unabhängig} \\[4mm]
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\begin{array}{rl}
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\text{Faltungssatz (diskret): } & P_Z(n) =
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\nsum_{k=0}^{n} P_X(k)P_Y(n-k) \\[2mm]
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\text{Charakteristische Funktion: } & \phi_Z(s) = \phi_X(s)
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\cdot \phi_Y(s)
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\end{array}
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\end{gather*}
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\end{frame}
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\begin{frame}
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@ -133,6 +133,7 @@
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\end{itemize}
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\end{itemize}
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\pause
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\begin{gather*}
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X \sim \text{Poisson}(\lambda)
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\end{gather*}
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@ -151,6 +152,8 @@
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\begin{frame}
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\frametitle{Zusammenfassung}
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\vspace*{-20mm}
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\begin{columns}[t]
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\column{\kitthreecolumns}
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\begin{greenblock}{Poisson-Verteilung}
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@ -289,12 +292,60 @@
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\subsection{Theorie Wiederholung}
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\begin{frame}
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\frametitle{Korrelationskoeffizient}
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\frametitle{Unabhängigkeit \& Korrelation}
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\vspace*{-10mm}
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\begin{itemize}
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\item Korrelation
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\item Unabhängige ZV (stetig)
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\begin{columns}
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\column{\kitthreecolumns}
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\begin{align*}
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X,Y \text{ unabhängig}
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\hspace{5mm} \Leftrightarrow \hspace{5mm}
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f_{X,Y}(x,y) = f_X(x)f_Y(y)
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\end{align*}
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\column{\kitthreecolumns}
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\begin{lightgrayhighlightbox}
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Erinnerung: Unabhängige Ereignisse
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\begin{align*}
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X,Y \text{ \normalfont unabhängig}
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\hspace{5mm} \Leftrightarrow \hspace{5mm}
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P(AB) = P(A)P(B)
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\end{align*}
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\vspace*{-13mm}
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\end{lightgrayhighlightbox}
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\end{columns}
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\pause
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\item Kovarianz
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\begin{columns}
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\column{\kitthreecolumns}
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\begin{align*}
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\text{cov}(X,Y) &= E\bigg( \big(X - E(X)\big) \big(Y
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- E(Y)\big) \bigg) \\
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&= E(XY) - E(X)E(Y)
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\end{align*}
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\column{\kitthreecolumns}
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\begin{lightgrayhighlightbox}
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Erinnerung: Varianz
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\begin{align*}
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V(X) = E\big( \left(X - E(X)\right)^2 \big) = E(X^2) - E^2(X)
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\end{align*}
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\vspace*{-13mm}
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\end{lightgrayhighlightbox}
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\end{columns}
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\item Korrelation
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\begin{align*}
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E(XY)
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\end{align*}
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\pause
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\item Korrelationskoeffizient
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\begin{align*}
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\rho_{XY} = \frac{\text{cov}(X,Y)}{\sqrt{V(X)V(Y)}}
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\hspace{25mm} \rho_{XY} = 0
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\hspace{2mm}\Leftrightarrow\hspace{2mm}
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E(XY) = E(X)E(Y)
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\end{align*}
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\end{itemize}
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\end{frame}
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@ -405,10 +456,204 @@
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\end{columns}
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\end{frame}
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\begin{frame}
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\frametitle{Unabhängigkeit vs. Korrelation}
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\vspace*{-15mm}
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\begin{itemize}
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\item Korrelation misst einen linearen Zusammenhang zwischen zwei ZV.\\
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Unabhängigkeit gibt an ob zwei ZV ``überhaupt zusammenhängen''
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\begin{align*}
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\hspace{5mm} X,Y \text{ unabhängig}
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\hspace{5mm}\Rightarrow\hspace{5mm}
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X,Y \text{ unkorreliert}
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\end{align*}
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\item Bei gemeinsam normalverteilten ZV gilt zusätzlich
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\begin{align*}
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\hspace{5mm} X,Y \text{ unkorreliert}
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\hspace{5mm}\Rightarrow\hspace{5mm}
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X,Y \text{ unabhängig}
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\end{align*}
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\vspace*{5mm}
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\pause
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\item Korrelation und Unabhängigkeit haben nichts mit den
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Einzelverteilungen zu tun. Sie sind ``eine Ebene höher''
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\begin{figure}[H]
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\centering
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\begin{subfigure}{0.32\textwidth}
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\begin{tikzpicture}[
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/pgfplots/scale only axis,
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/pgfplots/width=3.5cm,
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/pgfplots/height=3.5cm
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]
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\begin{axis}[
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name=main axis,
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view={0}{90},
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ticks=none,
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xlabel={$x$},ylabel={$y$},
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]
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\addplot3[
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surf, shader=interp,
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samples=40,
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domain=-3:3, y domain=-3:3
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]
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{1/(2*pi*sqrt(0.5)) * exp(-1/(2*(1 -
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sqrt(0.5))) * (x^2 -2*sqrt(0.5)*x*y + y^2) )};
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\end{axis}
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\node[below] at
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($(main axis.south west) + (-.5, -.5)$)
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{$f_{X,Y}(x,y)$};
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\begin{axis}[
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anchor=south west,
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at=(main axis.north west),
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height=2cm,
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ticks=none,
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ylabel={$f_X(x)$},
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samples=50,
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domain=-3:3,
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xmin=-3,xmax=3,
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]
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\addplot[line width=1pt] {1/sqrt(2*pi) *
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exp(-x^2/2)};
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\end{axis}
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\begin{axis}[
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anchor=north west,
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at=(main axis.north east),
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width=2cm,
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ticks=none,
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xlabel={$f_Y(y)$},
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samples=50,
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domain=-3:3,
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ymin=-3,ymax=3,
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]
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\addplot[line width=1pt] ( {1/sqrt(2*pi)
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* exp(-x^2/2)}, {x} );
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\end{axis}
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\end{tikzpicture}
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\end{subfigure}%
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\begin{subfigure}{0.32\textwidth}
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\begin{tikzpicture}[
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/pgfplots/scale only axis,
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/pgfplots/width=3.5cm,
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/pgfplots/height=3.5cm
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]
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\begin{axis}[
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name=main axis,
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view={0}{90},
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ticks=none,
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xlabel={$x$},ylabel={$y$},
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]
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\addplot3[
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surf, shader=interp,
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samples=40,
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domain=-3:3, y domain=-3:3
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]
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{1/(2*pi) * exp(-1/2 * (x^2 + y^2) )};
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\end{axis}
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\node[below] at
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($(main axis.south west) + (-.5, -.5)$)
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{$f_{X,Y}(x,y)$};
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\begin{axis}[
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anchor=south west,
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at=(main axis.north west),
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height=2cm,
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ticks=none,
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ylabel={$f_X(x)$},
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samples=50,
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domain=-3:3,
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xmin=-3,xmax=3,
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]
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\addplot[line width=1pt] {1/sqrt(2*pi) *
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exp(-x^2/2)};
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\end{axis}
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\begin{axis}[
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anchor=north west,
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at=(main axis.north east),
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width=2cm,
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ticks=none,
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xlabel={$f_Y(y)$},
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samples=50,
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domain=-3:3,
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ymin=-3,ymax=3,
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]
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\addplot[line width=1pt] ( {1/sqrt(2*pi)
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* exp(-x^2/2)}, {x} );
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\end{axis}
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\end{tikzpicture}
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\end{subfigure}%
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\begin{subfigure}{0.32\textwidth}
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\begin{tikzpicture}[
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/pgfplots/scale only axis,
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/pgfplots/width=3.5cm,
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/pgfplots/height=3.5cm
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]
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\begin{axis}[
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name=main axis,
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view={0}{90},
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ticks=none,
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xlabel={$x$},ylabel={$y$},
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]
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\addplot3[
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surf, shader=interp,
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samples=40,
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domain=-3:3, y domain=-3:3
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]
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{1/(2*pi*sqrt(0.5)) * exp(-1/(2*(1 -
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sqrt(0.5))) * (x^2 +2*sqrt(0.5)*x*y + y^2) )};
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\end{axis}
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\node[below] at
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($(main axis.south west) + (-.5, -.5)$)
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{$f_{X,Y}(x,y)$};
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\begin{axis}[
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anchor=south west,
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at=(main axis.north west),
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height=2cm,
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ticks=none,
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ylabel={$f_X(x)$},
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samples=50,
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domain=-3:3,
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xmin=-3,xmax=3,
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]
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\addplot[line width=1pt] {1/sqrt(2*pi) *
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exp(-x^2/2)};
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\end{axis}
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\begin{axis}[
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anchor=north west,
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at=(main axis.north east),
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width=2cm,
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ticks=none,
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xlabel={$f_Y(y)$},
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samples=50,
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domain=-3:3,
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ymin=-3,ymax=3,
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]
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\addplot[line width=1pt] ( {1/sqrt(2*pi)
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* exp(-x^2/2)}, {x} );
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\end{axis}
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\end{tikzpicture}
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\end{subfigure}
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\end{figure}
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\end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Zusammenfassung}
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\vspace*{-10mm}
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\vspace*{-20mm}
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\begin{columns}[t]
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\column{\kittwocolumns}
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@ -421,7 +666,7 @@
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\begin{greenblock}{Kovarianz}
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\vspace*{-6mm}
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\begin{gather*}
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\text{cov}(X,Y) = E(X\cdot Y) - E(X)E(Y)
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\text{cov}(X,Y) = E(X Y) - E(X)E(Y)
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\end{gather*}
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\end{greenblock}
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\begin{greenblock}{Randdichte}
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@ -595,50 +840,50 @@
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\begin{figure}[H]
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\centering
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% \begin{tikzpicture}
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% \begin{axis}[
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% view={20}{30},
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% xlabel=$x$, ylabel=$y$, zlabel={$f_{X,Y}(x,y)$},
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% xmin=0, xmax=1, ymin=0, ymax=1, zmin=0, zmax=2,
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% xtick={0,0.5,1},ytick={0,0.5,1},ztick={0,1,2},
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% point meta min=0, point meta max=2,
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% declare function={cutoff(\x) = 0.3/\x;},
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% legend,
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% ]
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% \addplot3[
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% surf, shader=interp,
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% samples=40,
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% domain=0:1, y domain=0:1
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% ] (
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% x,
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% {y * min(1, cutoff(x))},
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% {x + (y * min(1, cutoff(x)))}
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% );
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% \addlegendentry{$x\cdot y \le z$}
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%
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% \addplot3[
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% surf, shader=interp,
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% samples=40,
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% domain=0.3:1, y domain=0:1,
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% fill=gray,
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% draw=none,
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% point meta=1.1,
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% colormap name=cividis,
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% ] (
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% x,
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% {cutoff(x) + y*(1 - cutoff(x))},
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% {x + (cutoff(x) + y*(1 - cutoff(x)))}
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% );
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%
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% \addplot3[
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% mesh,
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% samples=15,
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% domain=0:1, y domain=0:1,
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% draw=black,
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% opacity=0.3
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% ] {x + y};
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% \end{axis}
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% \end{tikzpicture}
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\begin{tikzpicture}
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\begin{axis}[
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view={20}{30},
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xlabel=$x$, ylabel=$y$, zlabel={$f_{X,Y}(x,y)$},
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xmin=0, xmax=1, ymin=0, ymax=1, zmin=0, zmax=2,
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xtick={0,0.5,1},ytick={0,0.5,1},ztick={0,1,2},
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point meta min=0, point meta max=2,
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declare function={cutoff(\x) = 0.3/\x;},
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legend,
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]
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\addplot3[
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surf, shader=interp,
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samples=40,
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domain=0:1, y domain=0:1
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] (
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x,
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{y * min(1, cutoff(x))},
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{x + (y * min(1, cutoff(x)))}
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);
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\addlegendentry{$x\cdot y \le z$}
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\addplot3[
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surf, shader=interp,
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samples=40,
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domain=0.3:1, y domain=0:1,
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fill=gray,
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draw=none,
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point meta=1.1,
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colormap name=cividis,
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] (
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x,
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{cutoff(x) + y*(1 - cutoff(x))},
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{x + (cutoff(x) + y*(1 - cutoff(x)))}
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);
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\addplot3[
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mesh,
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samples=15,
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domain=0:1, y domain=0:1,
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draw=black,
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opacity=0.3
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] {x + y};
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\end{axis}
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\end{tikzpicture}
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\end{figure}
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\end{minipage}%
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\begin{minipage}{0.58\textwidth}
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