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Add formulas to summary slide of section 2

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Andreas Tsouchlos 2026-01-15 00:42:44 +01:00
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commit 587d894e5e
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src/2026-01-16/presentation.tex
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@ -202,6 +202,67 @@
% TODO:
\begin{frame}
\frametitle{Zusammenfassung}
\begin{columns}[t]
\column{\kittwocolumns}
\begin{greenblock}{Korrelationskoeffizient}
\vspace*{-6mm}
\begin{gather*}
\rho_{XY} = \frac{\text{cov}(X,Y)}{\sqrt{V(X)V(Y)}}
\end{gather*}
\end{greenblock}
\begin{greenblock}{Kovarianz}
\vspace*{-6mm}
\begin{gather*}
\text{cov}(X,Y) = E(X\cdot Y) - E(X)E(Y)
\end{gather*}
\end{greenblock}
\begin{greenblock}{Randdichte}
\vspace*{-6mm}
\begin{gather*}
f_X(x) = \int_{-\infty}^{\infty} f_{X,Y}(x,y) dy
\end{gather*}
\end{greenblock}
\column{\kitfourcolumns}
\begin{greenblock}{Transformationssatz}
\vspace*{-6mm}
\begin{gather*}
X = h_1(U,V), \hspace{5mm} Y = h_2(U,V) \\[2mm]
\mathcal{J} =
\begin{pmatrix}
\frac{\displaystyle \partial}{\displaystyle
\partial u}x & \frac{\displaystyle
\partial}{\displaystyle \partial v}x \\[2mm]
\frac{\displaystyle \partial}{\displaystyle
\partial u}y & \frac{\displaystyle
\partial}{\displaystyle \partial v}y
\end{pmatrix}
= \begin{pmatrix}
\frac{\displaystyle \partial}{\displaystyle
\partial u}h_1(u,v) & \frac{\displaystyle
\partial}{\displaystyle \partial v}h_1(u,v) \\[2mm]
\frac{\displaystyle \partial}{\displaystyle
\partial u}h_2(u,v) & \frac{\displaystyle
\partial}{\displaystyle \partial v}h_2(u,v)
\end{pmatrix} \\[3mm]
f_{U,V}(u,v) = \lvert
\text{det}(\mathcal{J}) \rvert
\cdot f_{X,Y} \big(h_1(u,v),h_2(u,v)\big)
\end{gather*}
\end{greenblock}
\begin{greenblock}{Erwartungswert \& Varianz}
\vspace*{-6mm}
\begin{align*}
V(X) &= E\big( (X - E(X))^2 \big) = E(X^2) - E^2(X) \\
E(X) &= \int_{-\infty}^{\infty} x f_X(x) dx \\
E(g(X)) &= \int_{-\infty}^{\infty} g(x) f_X(x) dx
\end{align*}
\end{greenblock}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Aufgabe}