From 307a47a7a761727fb9c431964497dcda89a51fa1 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Thu, 2 Apr 2026 07:37:50 +0200 Subject: [PATCH] [PS] Expected value done, Variance start --- .../ps-jh/parts/03_expected-value/02_cont.tex | 1 + .../parts/03_expected-value/03_properties.tex | 20 ++++++++++++++++++ .../03_expected-value/04_inequalities.tex | 6 ++++++ .../parts/03_expected-value/05_variance.tex | 2 ++ .../probability-and-statistics-cheatsheet.pdf | Bin 344420 -> 353126 bytes .../probability-and-statistics-cheatsheet.tex | 3 +++ 6 files changed, 32 insertions(+) create mode 100644 semester4/ps/ps-jh/parts/03_expected-value/03_properties.tex create mode 100644 semester4/ps/ps-jh/parts/03_expected-value/04_inequalities.tex create mode 100644 semester4/ps/ps-jh/parts/03_expected-value/05_variance.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 index 9b68fd5..e6e937a 100644 --- a/semester4/ps/ps-jh/parts/03_expected-value/02_cont.tex +++ b/semester4/ps/ps-jh/parts/03_expected-value/02_cont.tex @@ -5,6 +5,7 @@ \subsubsection{Beispiele} % TODO: Consider if need derivation of them here and prev section as well +% TODO: Also add the ones proven in exercises \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 $\cX \sim \cU([a, b])$, $a < b$: $\E[\cX] = \frac{a + b}{2}$ diff --git a/semester4/ps/ps-jh/parts/03_expected-value/03_properties.tex b/semester4/ps/ps-jh/parts/03_expected-value/03_properties.tex new file mode 100644 index 0000000..6c84ba8 --- /dev/null +++ b/semester4/ps/ps-jh/parts/03_expected-value/03_properties.tex @@ -0,0 +1,20 @@ +\subsection{Eigenschaften des Erwartungswerts} +\shorttheorem[Linearität] Falls $\E[\cX]$ und $\E[\cY]$ wohldefiniert:\\ +$\E[\lambda \cX] = \lambda \E[\cX]$ und $\E[\cX + \cY] = \E[\cX] + \E[\cY]$ + +\shortremark Z.V $\cX_k$ und $\lambda_k$: $\E\left[ \sum_{k = 1}^{n} \lambda_k \cX_k \right] = \sum_{k = 1}^{n} \lambda_k \E[\cX_k]$ + +\shorttheorem[Monotonie] Sei $\cX \leq \cY$ mit $\E$ wohldef.: $\E[\cX] \leq \E[\cY]$ + +\shorttheorem $\cX, \cY$ unabh., dann $\E[\cX \cY] = \E[\cX] \E[\cY]$ + +\shorttheorem $\cX_k$ alle unabhängig mit $\E[\cX_k]$ endlich. Dann gilt\\ +$\E\left[ \prod_{k = 1}^n \cX_k \right] = \prod_{k = 1}^n \E[\cX_k]$ + +\shorttheorem $f : \R \rightarrow \R_+$ mit $\int_{-\8}^{\8} f(x) \dx x = 1$. Dann ist äquivalent: +\bi{(1)} $\cX$ stetig mit Dichte $f$ und \bi{(2)} für jede stückweise stetige Abb. $\varphi : \R \rightarrow \R$ gilt $\E[\varphi(\cX)] = \int_{-\8}^{\8} \varphi(x) f(x) \dx x$ + +\shorttheorem äquivalent: \bi{1} $\cX, \cY$ unabhängig, für alle $\varphi, \psi : \R \rightarrow \R$: $\E[\varphi(\cX) \psi(\cX)] = \E[\varphi(\cX)] \E[\psi]$ + +\shorttheorem äquivalent: \bi{(1)} $\cX_i$ unabhängig,\\ +\bi{(2)} $\forall \varphi_i$: $\E[\varphi_1(\cX_1) \cdots \varphi_n(\cX_n)] = \E[\varphi_1(\cX_1)] \cdots \E[\varphi_n(\cX_n)]$ diff --git a/semester4/ps/ps-jh/parts/03_expected-value/04_inequalities.tex b/semester4/ps/ps-jh/parts/03_expected-value/04_inequalities.tex new file mode 100644 index 0000000..1a39fea --- /dev/null +++ b/semester4/ps/ps-jh/parts/03_expected-value/04_inequalities.tex @@ -0,0 +1,6 @@ +\subsection{Ungleichungen} +\shorttheorem[Markow] $\cX$ n.-neg., $g : \cX(\Omega) \rightarrow [0, \8)$. $\forall c \in \R$ mit $g(c) > 0$ gilt $\P[\cX \geq c] = \frac{\E[g(\cX)]}{g(c)}$ + +\shorttheorem[Jensensche] $\varphi: \R \rightarrow \R$ konvex, und falls $\E[\varphi(\cX)]$ und $\E[\cX]$ wohldefiniert: $\varphi(\E[\cX]) \leq \E[\varphi(\cX)]$ + +\shorttheorem[Dreieck] $\varphi(x) = |x|$, dann $|\E[\cX]| \leq \E[|\cX|]$. $\varphi(x) = x^2$, dann $\E[|\cX|] \leq \sqrt{\E[\cX^2]}$ diff --git a/semester4/ps/ps-jh/parts/03_expected-value/05_variance.tex b/semester4/ps/ps-jh/parts/03_expected-value/05_variance.tex new file mode 100644 index 0000000..f82fb7b --- /dev/null +++ b/semester4/ps/ps-jh/parts/03_expected-value/05_variance.tex @@ -0,0 +1,2 @@ +\subsection{Varianz} +\shortdefinition $\cX$ mit $\E[\cX^2] < \8$ diff --git a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf index 425f9f2b3c1829f8d57318134ed1031d60d74696..35ba2656abdb448599acb17fd0fbf539e2b5bf6f 100644 GIT binary patch delta 119864 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