diff --git a/src/2025-12-19/presentation.tex b/src/2025-12-19/presentation.tex index 26a53fe..436f23b 100644 --- a/src/2025-12-19/presentation.tex +++ b/src/2025-12-19/presentation.tex @@ -81,7 +81,7 @@ \subsection{Theorie Wiederholung} \begin{frame} - \frametitle{Stetige Zufallsvariablen} + \frametitle{Stetige Zufallsvariablen I} \vspace*{-10mm} @@ -103,7 +103,8 @@ \begin{gather*} F_X(x) = P(X \le x) \end{gather*} - \pause\vspace{-10mm}\begin{gather*} + \pause\vspace{-10mm} + \begin{gather*} \text{Eigenschaften:} \\[3mm] \lim_{x\rightarrow -\infty} F_X(x) = 0 \\ \lim_{x\rightarrow +\infty} F_X(x) = 1 \\ @@ -118,7 +119,8 @@ \begin{gather*} F_X(x) = \int_{-\infty}^{x} f_X(u) du \end{gather*} - \pause\vspace{-10mm}\begin{gather*} + \pause\vspace{-10mm} + \begin{gather*} \text{Eigenschaften:} \\[3mm] f_X(x) \ge 0 \\ \int_{-\infty}^{\infty} f_X(x) dx = 1 @@ -130,9 +132,36 @@ % TODO: Write this % TODO: P(a < X < b) = F_X(b) - F_X(a) % TODO: E(X_cont) = int(...) vs E(X_disc) = sum(...) -% TODO: V(X) = ... (preparation for use of sigma^2 in part two) \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}