From ea57bd5269a434c2049415e247c47858249adc11 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Thu, 26 Mar 2026 11:16:15 +0100 Subject: [PATCH] [PS] Expected Value: Continuous random variables --- .../{00_cont.tex => 00_intro.tex} | 0 .../ps-jh/parts/03_expected-value/02_cont.tex | 15 +++++++++++++++ .../probability-and-statistics-cheatsheet.pdf | Bin 343792 -> 344966 bytes .../probability-and-statistics-cheatsheet.tex | 3 ++- 4 files changed, 17 insertions(+), 1 deletion(-) rename semester4/ps/ps-jh/parts/03_expected-value/{00_cont.tex => 00_intro.tex} (100%) create mode 100644 semester4/ps/ps-jh/parts/03_expected-value/02_cont.tex 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 3e6ab4b42227f3fd67917952c65fb036e1b68bd6..1c64b96203e8a219e0b8a702403ae03fef4eb815 100644 GIT binary patch delta 71507 zcmV)6K*+!F{S=0W6|hDLldvc>12#A~lVJoXf9+XYZxcBbe$TJ)BS>uG>)4j6No%Suam2+6cWO%s$0m;LqoouTp66HhWB2`vz*vCsH&&UY?8zSJtY0t6@okpT|G?`@%?IuHQc*|Ipd5aIuEEixVn3WRW8Z|yL!>rC|5^4{o 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