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Andreas Tsouchlos 2025-12-16 17:42:26 +01:00
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commit c0992e9690
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src/2025-12-19/presentation.tex
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@ -81,7 +81,7 @@
\subsection{Theorie Wiederholung} \subsection{Theorie Wiederholung}
\begin{frame} \begin{frame}
\frametitle{Stetige Zufallsvariablen} \frametitle{Stetige Zufallsvariablen I}
\vspace*{-10mm} \vspace*{-10mm}
@ -103,7 +103,8 @@
\begin{gather*} \begin{gather*}
F_X(x) = P(X \le x) F_X(x) = P(X \le x)
\end{gather*} \end{gather*}
\pause\vspace{-10mm}\begin{gather*} \pause\vspace{-10mm}
\begin{gather*}
\text{Eigenschaften:} \\[3mm] \text{Eigenschaften:} \\[3mm]
\lim_{x\rightarrow -\infty} F_X(x) = 0 \\ \lim_{x\rightarrow -\infty} F_X(x) = 0 \\
\lim_{x\rightarrow +\infty} F_X(x) = 1 \\ \lim_{x\rightarrow +\infty} F_X(x) = 1 \\
@ -118,7 +119,8 @@
\begin{gather*} \begin{gather*}
F_X(x) = \int_{-\infty}^{x} f_X(u) du F_X(x) = \int_{-\infty}^{x} f_X(u) du
\end{gather*} \end{gather*}
\pause\vspace{-10mm}\begin{gather*} \pause\vspace{-10mm}
\begin{gather*}
\text{Eigenschaften:} \\[3mm] \text{Eigenschaften:} \\[3mm]
f_X(x) \ge 0 \\ f_X(x) \ge 0 \\
\int_{-\infty}^{\infty} f_X(x) dx = 1 \int_{-\infty}^{\infty} f_X(x) dx = 1
@ -130,9 +132,36 @@
% TODO: Write this % TODO: Write this
% TODO: P(a < X < b) = F_X(b) - F_X(a) % TODO: P(a < X < b) = F_X(b) - F_X(a)
% TODO: E(X_cont) = int(...) vs E(X_disc) = sum(...) % TODO: E(X_cont) = int(...) vs E(X_disc) = sum(...)
% TODO: V(X) = ... (preparation for use of sigma^2 in part two)
\begin{frame} \begin{frame}
\frametitle{TODO} \frametitle{Stetige Zufallsvariablen II}
\begin{columns}
\column{\kitfourcolumns}
\centering
\begin{itemize}
\item Kenngrößen
\begin{align*}
\begin{array}{rlr}
\text{Erwartungswert: } \hspace{5mm} & E(X) =
\displaystyle\int_{-\infty}^{\infty} x f_X(x) dx
& \hspace{5mm} \big( = \mu \big) \\
\text{Varianz: } \hspace{5mm} & V(X) = E\mleft(
\mleft( X - E(X) \mright)^2 \mright) \\
\text{Standardabweichung: } \hspace{5mm} &
\sqrt{V(X)} & \hspace{5mm} \big( = \sigma \big)
\end{array}
\end{align*}
\end{itemize}
\column{\kittwocolumns}
\centering
\begin{lightgrayhighlightbox}
Erinnerung
\begin{align*}
\text{\normalfont Erwartungswert: }& E(X) = \sum_{n=1}^{\infty} x_n P_X(x) \\
\text{\normalfont Varianz: }& V(X) = E\mleft( \mleft( X - E(X) \mright)^2 \mright)
\end{align*}
\end{lightgrayhighlightbox}
\end{columns}
\end{frame} \end{frame}