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Finish theory for part 1

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Andreas Tsouchlos 2025-12-16 23:13:53 +01:00
parent c0992e9690
commit aae0aae77b
1 changed files with 33 additions and 42 deletions
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src/2025-12-19/presentation.tex
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@ -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}
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