diff --git a/electives/others/amr/parts/02_Sensors-Actuators/03_cameras.tex b/electives/others/amr/parts/02_Sensors-Actuators/03_cameras.tex new file mode 100644 index 0000000..adbf8da --- /dev/null +++ b/electives/others/amr/parts/02_Sensors-Actuators/03_cameras.tex @@ -0,0 +1,9 @@ +\shortdefinition[Pinhole projection] +$\begin{bmatrix} + u \\ v + \end{bmatrix} + = \frac{f}{z} + \begin{bmatrix} + x \\ y + \end{bmatrix}$ + with $f$ the distance to the lens and $z$ the full distance diff --git a/semester4/ps/ps-jh/parts/03_expected-value/00_cont.tex b/semester4/ps/ps-jh/parts/03_expected-value/00_cont.tex new file mode 100644 index 0000000..ec9f7a2 --- /dev/null +++ b/semester4/ps/ps-jh/parts/03_expected-value/00_cont.tex @@ -0,0 +1,13 @@ +\subsection{Allgemeiner Erwartungswert} +\shortdefinition Für $\cX : \Omega \rightarrow \R_+$, $\E[\cX] = \int_{0}^{\8} (1 - F_\cX(x)) \dx x$ + +\shortremark $\E[\cX]$ immer definiert und endlich oder unendlich + +\shorttheorem $\cX$ n.-neg. Dann: $\E[\cX] \geq 0$. $=$, wenn $\cX = 0$ fast sicher + +\shortdefinition $\E[\cX] = \E[\cX_+] - \E[\cX_-]$ mit $\cX_-$ auch n.-neg. + +\shortremark $|\cX| = \cX_+ + \cX_-$. Für $\cX \geq 0$ ist $\E[\cX]$ immer definiert. +Falls $\cX$ kein konst. Vorzeichen, $\E[\cX]$ undef. + +\shortremark $\E[\cX] = \int_{0}^{\8} (1 - F_\cX(x)) \dx x - \int_{-\8}^{0} F_\cX(x)$ diff --git a/semester4/ps/ps-jh/parts/03_expected-value/01_disc.tex b/semester4/ps/ps-jh/parts/03_expected-value/01_disc.tex new file mode 100644 index 0000000..5b302eb --- /dev/null +++ b/semester4/ps/ps-jh/parts/03_expected-value/01_disc.tex @@ -0,0 +1,14 @@ +\subsection{Diskrete Zufallsvariablen} +\shorttheorem Für $\cX$ mit Werten fast sicher in $W$: +\[ + \E[\cX] = \sum_{x \in W} x \cdot \P[\cX = x] = \sum_{x \in W} x \cdot p_\cX(x) +\] + +\shortremark $\E[\cX]$ wohldefiniert falls $(x \cdot p_\cX(x))_{x \in W}$ abs. konv. + +\subsubsection{Beispiele} +\begin{itemize} + \item $\cX \sim \text{Ber}(p)$: $\E[\cX] = p$ + \item $\cX \sim \text{Bin}(n, p)$: $\E[\cX] = np$ + \item $\cX \sim \text{Poisson}(\lambda)$: $\E[\cX] = \lambda$ +\end{itemize} diff --git a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf index 120ee4d..b6bc250 100644 Binary files a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf and b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf differ diff --git a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex index 6ec4a5b..3d1ec60 100644 --- a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex +++ b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex @@ -59,5 +59,11 @@ \input{parts/02_discrete-continuous-rv/05_cont-distributions/03_normal.tex} % \input{parts/02_discrete-continuous-rv/} +\newsectionNoPB +\section{Erwartungswert} +\input{parts/03_expected-value/00_cont.tex} +\input{parts/03_expected-value/01_disc.tex} +% \input{parts/03_expected-value/} + \end{document}