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tut4: Started adding theory for exercise 1

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Andreas Tsouchlos 2025-12-16 00:18:12 +01:00
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src/2025-12-19/presentation.tex
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@ -81,7 +81,90 @@
\subsection{Theorie Wiederholung}
\begin{frame}
\frametitle{sasdf}
\frametitle{Stetige Zufallsvariablen}
\vspace*{-10mm}
\begin{lightgrayhighlightbox}
Erinnerung: Diskrete Zufallsvariablen
\begin{align*}
\text{\normalfont Verteilung: }& P_X(x) = P(X = x) \\
\text{\normalfont Verteilungsfunktion: }& F_X(x) = P(X \le x) =
\sum_{n: x_n \le y} P_X(x)
\end{align*}
\vspace{-10mm}
\end{lightgrayhighlightbox}
\begin{columns}[t]
\pause\column{\kitthreecolumns}
\centering
\begin{itemize}
\item Verteilungsfunktion $F_X(x)$ einer stetiger ZV
\begin{gather*}
F_X(x) = P(X \le x) \\[5mm]
\text{Eigenschaften:} \\[3mm]
\lim_{x\rightarrow -\infty} F_X(x) = 0 \\
\lim_{x\rightarrow +\infty} F_X(x) = 1 \\
x_1 \le x_2 \Rightarrow F_X(x_1) \le F_X(x_2)\\
\lim_{h\rightarrow 0^+} F_X(x + h) = F_X(x)
\end{gather*}
\end{itemize}
\pause\column{\kitthreecolumns}
\centering
\begin{itemize}
\item Wahrscheinlichkeitsdichte $f_X(x)$ einer stetiger ZV
\begin{gather*}
F_X(x) = \int_{-\infty}^{x} f_X(u) du \\[5mm]
\text{Eigenschaften:} \\[3mm]
f_X(x) \ge 0 \\
\int_{-\infty}^{\infty} f_X(x) dx = 1
\end{gather*}
\end{itemize}
\end{columns}
\end{frame}
% TODO: Write this
\begin{frame}
\frametitle{TODO}
\end{frame}
\begin{frame}
\frametitle{Zusammenfassung}
\begin{columns}[c]
\column{\kitthreecolumns}
\centering
\begin{greenblock}{Verteilungsfunktion (kontinuierlich)}
\vspace*{-6mm}
\begin{gather*}
F_X(x) = P(X \le x)\\[8mm]
\lim_{x\rightarrow -\infty} F_X(x) = 0 \\
\lim_{x\rightarrow +\infty} F_X(x) = 1 \\
x_1 \le x_2 \Rightarrow F_X(x_1) \le F_X(x_2)\\
\lim_{h\rightarrow 0^+} F_X(x + h) = F_X(x)
\end{gather*}
\end{greenblock}
\column{\kitthreecolumns}
\centering
\begin{greenblock}{Wahrscheinlichkeitsdichte \phantom{()}}
\vspace*{-6mm}
\begin{gather*}
F_X(x) = \int_{-\infty}^{x} f_X(u) du \\[5mm]
f_X(x) \ge 0 \\
\int_{-\infty}^{\infty} f_X(x) dx = 1
\end{gather*}
\end{greenblock}
%TODO: Rename this
\begin{greenblock}{TODO}
\vspace*{-6mm}
\begin{gather*}
P(a < X \le b) = F_X(b) - F_X(a) \\[2mm]
E(X) = \int_{-\infty}^{x} u f_X(u) du
\end{gather*}
\end{greenblock}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%