diff --git a/semester4/ps/ps-jh/parts/03_expected-value/00_cont.tex b/semester4/ps/ps-jh/parts/03_expected-value/00_intro.tex similarity index 100% rename from semester4/ps/ps-jh/parts/03_expected-value/00_cont.tex rename to semester4/ps/ps-jh/parts/03_expected-value/00_intro.tex diff --git a/semester4/ps/ps-jh/parts/03_expected-value/02_cont.tex b/semester4/ps/ps-jh/parts/03_expected-value/02_cont.tex new file mode 100644 index 0000000..ce2fbf6 --- /dev/null +++ b/semester4/ps/ps-jh/parts/03_expected-value/02_cont.tex @@ -0,0 +1,15 @@ +\subsection{Stetige Zufallsvariablen} +\shortdefinition $\cX$ stetig $\E[\cX] = \int_{-\8}^{\8} x _\cX(x) \dx x$ + +\shorttheorem $\E[\varphi(\cX)] = \int_{-\8}^{\8} \varphi(x) f_\cX(x) \dx x$, falls int. wohldefiniert + +\subsubsection{Beispiele} +% TODO: Consider if need derivation of them here and prev section as well +\shortlemma[Int über gauss. Glockenk.] $\int_{-\8}^{\8} e^{\frac{-x^2}{2\sigma^2}} \dx x = \sqrt{2 \pi \sigma^2}$ +\begin{itemize} + \item $\cX \sim \cU([a, b])$, $a < b$: $\E[\cX] = \frac{a + b}{2}$ + \item $\cX \sim \text{Exp}(\lambda)$, $\lambda > 0$: $\E[\cX] = \frac{1}{\lambda}$ + \item $\cX \sim \cU(\mu, \sigma^2)$, $z = x - \mu$, $\dx z = \dx x$: $\E[\cX] = \mu$ + \item $\cX \sim \text{Cauchy}(x_0, \gamma)$: Existiert nicht (Int. $\8$)\\ + $\E[\cX_+] = \E[\cX_-] = \8$, Median: $0$ +\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 3e6ab4b..1c64b96 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 3d1ec60..d504513 100644 --- a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex +++ b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex @@ -61,8 +61,9 @@ \newsectionNoPB \section{Erwartungswert} -\input{parts/03_expected-value/00_cont.tex} +\input{parts/03_expected-value/00_intro.tex} \input{parts/03_expected-value/01_disc.tex} +\input{parts/03_expected-value/02_cont.tex} % \input{parts/03_expected-value/}