From f71cb9841f6d60bd80b16ccdc1c20e823e0cc5e0 Mon Sep 17 00:00:00 2001 From: RobinB27 Date: Thu, 26 Mar 2026 13:02:18 +0100 Subject: [PATCH] [PS] Expected Value --- semester4/ps/ps-rb/main.pdf | Bin 265417 -> 280795 bytes semester4/ps/ps-rb/main.tex | 5 +++ semester4/ps/ps-rb/parts/01_variables.tex | 18 +++++++- semester4/ps/ps-rb/parts/02_expectation.tex | 45 ++++++++++++++++++++ semester4/ps/ps-rb/util/helpers.tex | 1 + 5 files changed, 68 insertions(+), 1 deletion(-) create mode 100644 semester4/ps/ps-rb/parts/02_expectation.tex diff --git a/semester4/ps/ps-rb/main.pdf b/semester4/ps/ps-rb/main.pdf index d56e5b7ca9242b5b9c9acfbd3b9c960f1cf4af7c..583031bd08ab0d853d8268da35824faa27a24971 100644 GIT binary patch delta 157583 zcmV(lK=i-KnGoBN5f3F$L`E$!E;R}z(A34)3YT!J0~E0%QxX9;myx#v6SImE8yEpN zmyx#v69hFfH#C zM_)YCdKRnbl$*Ucn;97@W6ey-N*FtPaXS0?NFTp^@n`n$Mdj;%FTLIP*V7k|{`E+r znV3oH4*grJtLVe*_0=Q#T8YYG45_LR-Rx#DJA3p$!!AcBs!XL!M3)X<`Ff(?EY6y# 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Tassion} \section{Wahrscheinlichkeitsräume} \input{parts/00_intro.tex} + \newpage \section{Zufallsvariablen} \input{parts/01_variables.tex} +\newpage +\section{Erwartungswert} +\input{parts/02_expectation.tex} + \end{document} diff --git a/semester4/ps/ps-rb/parts/01_variables.tex b/semester4/ps/ps-rb/parts/01_variables.tex index 24b6848..d3f5aaf 100644 --- a/semester4/ps/ps-rb/parts/01_variables.tex +++ b/semester4/ps/ps-rb/parts/01_variables.tex @@ -229,4 +229,20 @@ $$ \end{cases} $$ -\definition \textbf{Exponentialverteilung} +\definition \textbf{Exponentialverteilung} $T \sim \text{exp}(\lambda)$ +$$ + f_T(x) = \begin{cases} + \lambda e^{-\lambda x} & x \geq 0 \\ + 0 & x < 0 + \end{cases} +$$ +\subtext{$\lambda > 0$} + +\lemma \textbf{Eigenschaften von} $\text{exp}$ + +\definition \textbf{Normalverteilung} $X \sim \mathcal{N}(m, \sigma^2)$ +$$ + f_X(x) = \frac{1}{ \sqrt{2\pi\sigma^2} } e^{ -\frac{(x-m)^2}{2\sigma^2} } +$$ + +\lemma \textbf{Eigenschaften von} $\mathcal{N}$ diff --git a/semester4/ps/ps-rb/parts/02_expectation.tex b/semester4/ps/ps-rb/parts/02_expectation.tex new file mode 100644 index 0000000..1ff199f --- /dev/null +++ b/semester4/ps/ps-rb/parts/02_expectation.tex @@ -0,0 +1,45 @@ +\definition \textbf{Erwartungswert} (nicht-negativ) +$$ + \E[X] = \int_{0}^{\infty} \biggl( 1-F_X(x) \biggr)\ dx +$$ +\subtext{$X: \Sigma \to \R,\quad X(\omega) \geq 0\ \forall \omega \in \Omega$} + +{\scriptsize + \remark $\E[X]$ kann unendlich sein +} + +\theorem $\forall \omega \in \Omega: X(\omega) \geq 0 \implies \E[X] \geq 0$\\ +\subtext{Gleichheit: $\E[X] = 0 \iff X=0$, fast sicher} + +\definition \textbf{Erwartungswert} +$$ + \E[X] = \E[X_+] - \E[X_-] \quad\text{falls}\quad \E[|X|]<\infty +$$ +{\scriptsize + \remark $X$ kein konst. Vorzeichen, nicht $\E[|X|] < \infty $: $\E[X]$ undefiniert +} + +\subsection{Diskreter Erwartungswert} + +\theorem \textbf{Diskreter Erwartungswert} +$$ + \E\bigl[ \phi(X) \bigr] = \sum_{w \in W} \phi(x) \cdot \P[X = x] +$$ +\subtext{$X: \Omega \to \R,\quad W \cleq \N,\quad \phi: \R \to \R$} + +\begin{center} + \begin{tabular}{l|l} + $\text{Ber}(p)$ & $\E[X] = p$ \\ + $\text{Poisson}(\lambda)$ & $\E[X] = \lambda$ \\ + $\text{Bin}(n,p)$ & $\E[X] = n\cdot p$ \\ + $\mathbb{I}_A$ & $\E[\mathbb{I}_A] = \P[A]$ \\ + \end{tabular} +\end{center} + +\subsection{Stetiger Erwartungswert} + +\definition \textbf{Erwartungswert} (stetig) +$$ + \E[X] = \int_{-\infty}^{\infty} x \cdot f(x)\ dx +$$ +\subtext{$X: \Omega \to \R,\quad f(x) \text{ Dichtefunktion}$} diff --git a/semester4/ps/ps-rb/util/helpers.tex b/semester4/ps/ps-rb/util/helpers.tex index 39bfe45..4f998c9 100644 --- a/semester4/ps/ps-rb/util/helpers.tex +++ b/semester4/ps/ps-rb/util/helpers.tex @@ -59,6 +59,7 @@ \def \P{\mathbb{P}} \def \F{\mathcal{F}} +\def \E{\mathbb{E}} % Titles \def \definition{\colorbox{lightgray}{Def} }