diff --git a/src/2025-12-19/presentation.tex b/src/2025-12-19/presentation.tex index 436f23b..4c53b8f 100644 --- a/src/2025-12-19/presentation.tex +++ b/src/2025-12-19/presentation.tex @@ -103,14 +103,6 @@ \begin{gather*} F_X(x) = P(X \le x) \end{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 \\ - 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 @@ -119,50 +111,58 @@ \begin{gather*} F_X(x) = \int_{-\infty}^{x} f_X(u) du \end{gather*} - \pause\vspace{-10mm} - \begin{gather*} - \text{Eigenschaften:} \\[3mm] - f_X(x) \ge 0 \\ - \int_{-\infty}^{\infty} f_X(x) dx = 1 - \end{gather*} \end{itemize} \end{columns} + \begin{columns}[t] + \pause \column{\kitthreecolumns} + \centering + \begin{gather*} + \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)\\ + F_X(x+) = \lim_{h\rightarrow 0^+} F_X (x+h) = F_X(x) + \end{gather*} + \pause \column{\kitthreecolumns} + \centering + \begin{gather*} + \text{Eigenschaften:} \\[3mm] + f_X(x) \ge 0 \\ + \int_{-\infty}^{\infty} f_X(x) dx = 1 + \end{gather*} + \end{columns} \end{frame} -% TODO: Write this -% TODO: P(a < X < b) = F_X(b) - F_X(a) -% TODO: E(X_cont) = int(...) vs E(X_disc) = sum(...) \begin{frame} \frametitle{Stetige Zufallsvariablen II} - \begin{columns} - \column{\kitfourcolumns} - \centering + \begin{minipage}{0.6\textwidth} \begin{itemize} - \item Kenngrößen + \item Wichtige 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) \\ + & \hspace{5mm} \big( = \mu \big) \\[3mm] \text{Varianz: } \hspace{5mm} & V(X) = E\mleft( - \mleft( X - E(X) \mright)^2 \mright) \\ + \mleft( X - E(X) \mright)^2 \mright) \\[3mm] \text{Standardabweichung: } \hspace{5mm} & \sqrt{V(X)} & \hspace{5mm} \big( = \sigma \big) \end{array} \end{align*} \end{itemize} - \column{\kittwocolumns} - \centering + \end{minipage} + \begin{minipage}{0.38\textwidth} \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) + \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{minipage} \end{frame} \begin{frame} @@ -174,11 +174,12 @@ \begin{greenblock}{Verteilungsfunktion (kontinuierlich)} \vspace*{-6mm} \begin{gather*} - F_X(x) = P(X \le x)\\[8mm] + F_X(x) = P(X \le x)\\[4mm] + P(a < X \le b) = F_X(b) - F_X(a) \\[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) + F_X(x+) = \lim_{h\rightarrow 0^+} F_X (x+h) = F_X(x) \end{gather*} \end{greenblock} \column{\kitthreecolumns} @@ -191,15 +192,6 @@ \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} @@ -339,7 +331,6 @@ \begin{gather*} x_1 \le x_2 \Rightarrow F_X(x_1) \le F_X(x_2) \\ F_X(x+) = \lim_{h\rightarrow 0^+} F_X (x+h) = F_X(x) - \hspace{5mm}\forall x\in \mathbb{R} \end{gather*} \column{\kitonecolumn} \end{columns} @@ -386,7 +377,7 @@ = P(1 < X \le 2) = F_X(2) - F_X(1) = e^{-a} - e^{-4a} \end{gather*} \end{enumerate} - % tex-fmt: off + % tex-fmt: on \end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%