From 5cb65871b7ae7286031c2cb4dfb8bfe87db6ecc7 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Thu, 2 Apr 2026 09:28:53 +0200 Subject: [PATCH] [PS] Variance and covariance complete --- .../parts/03_expected-value/05_variance.tex | 25 ++++++++++- .../parts/03_expected-value/06_covariance.tex | 40 ++++++++++++++++++ .../probability-and-statistics-cheatsheet.pdf | Bin 353126 -> 365597 bytes .../probability-and-statistics-cheatsheet.tex | 5 +++ 4 files changed, 69 insertions(+), 1 deletion(-) create mode 100644 semester4/ps/ps-jh/parts/03_expected-value/06_covariance.tex 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 index f82fb7b..8948baf 100644 --- a/semester4/ps/ps-jh/parts/03_expected-value/05_variance.tex +++ b/semester4/ps/ps-jh/parts/03_expected-value/05_variance.tex @@ -1,2 +1,25 @@ \subsection{Varianz} -\shortdefinition $\cX$ mit $\E[\cX^2] < \8$ +\shortdefinition $\cX$ mit $\E[\cX^2] < \8$, $\V[\cX] = \E[(\cX - \E[\cX])^2]$ + +\shortdefinition[Standardabweichung] $\sigma(\cX) = \sqrt{\V[\cX]}$ + +\shortremark $\V[\cX] = \E[\cX^2] - \E[\cX]^2$ + +\shortexample $\cX$ determ. Z.V (= konst) mit Wert $a$, also $\cX = a1_\Omega$. Dann: $\E[\cX] = a \E[1_\Omega] = a \P[\Omega] = a$ und\\ +$\V[\cX] = \E[\cX^2] - \E[\cX]^2 = a^2\E[1_\Omega] - a^2 = 0$ + +% Task 4.42 (needs proof?) +\shortremark $\E[\cX] < \8$, dann $\V[\cX] \geq 0$ mit $=$ g.d.w. $\cX$ konst; zudem $\V[a \cX] = a^2 \V[\cX]$ und $\V[\cX + a] = \V[\cX]$ + +\shortproposition $\cX_k$ paarw. unabh. $\V\left[ \sum_{k = 1}^{n} \cX_k \right] = \sum_{k = 1}^{n} \V[\cX_k]$ + +\shortexample Varianz von bekannten Verteilungen +\begin{itemize} + \item $\cX \sim \text{Ber}(p)$, $\V[\cX] = p (1 - p)$ + \item $\cX \sim \text{Bin}(n, p)$, $\V[\cX] = n p (1 - p)$ + \item $\cX \sim \text{Poisson}(\lambda)$, $\V[\cX] = \lambda = \E[\cX]$ + \item $\cX \sim \cU([a, b])$, $\V[\cX] = \frac{(b - a)^2}{12}$ + \item $\cX \sim \cN(\mu, \sigma^2)$, $\V[\cX] = \sigma^2$ +\end{itemize} + +\shortcorollary[Cheb.] $\V[\cY]$ end. $\forall c > 0$ gilt: $\P[|\cY - \E[\cY]| \geq c] \leq \frac{\V[\cY]}{c^2}$ diff --git a/semester4/ps/ps-jh/parts/03_expected-value/06_covariance.tex b/semester4/ps/ps-jh/parts/03_expected-value/06_covariance.tex new file mode 100644 index 0000000..03943d7 --- /dev/null +++ b/semester4/ps/ps-jh/parts/03_expected-value/06_covariance.tex @@ -0,0 +1,40 @@ +\newpage +\subsection{Kovarianz} +\shortdefinition $\cov(\cX, \cY) = \E[(\cX - \E[\cX])(\cY - \E[\cY])]$ + +\shortremark $\cov(\cX, \cY) = \E[\cX \cY] - \E[\cX] \E[\cY]$ + +\shortremark $\cov(\cX, \cX) = \V[\cX]$ + +\shortremark $\cov(\cX, \cY) = 0 \Longrightarrow \cX, \cY$ unabh. ($\Leftarrow$ impl. falsch!) + +\shortremark $\cX, \cY$ unabh. $\Longleftrightarrow \elementstack{\forall \varphi, \psi \text{stückweise stetig, beschränkt gilt}}{\cov(\varphi(\cX), \psi(\cY)) = 0}$ + +\shortremark Folgende Terminologie (neg. korr. = antikorreliert): +\begin{itemize} + \item Wenn $\cov(\cX, \cY) > 0$, dann: $\cX$, $\cY$ \bi{positiv korreliert} + \item Wenn $\cov(\cX, \cY) = 0$, dann: $\cX$, $\cY$ \bi{unkorreliert} + \item Wenn $\cov(\cX, \cY) < 0$, dann: $\cX$, $\cY$ \bi{negativ korreliert} +\end{itemize} + +\shortexample $\cX, \cY$ unkorreliert $\centernot\implies \cX, \cY$ unabhängig + +\shortremark Eigenschaften der Kovarianz (alle $a, \ldots \in \R$): +\begin{itemize} + \item \bi{Positive Semidefinitheit}: $\cov(\cX, \cX) \geq 0$ + \item \bi{Symmetrie}: $\cov(\cX, \cY) = \cov(\cY, \cX)$ + \item \bi{Bilin.}: $\cov(a\cX + b, c\cY + d) = ac \cov(\cX, \cY)$ und\\ + $\cov(\cX, (e \cY + f) + (g \cZ + h))\! =\! e \cov(\cX, \cY) + g \cov(\cX, \cZ)$ +\end{itemize} + +\shortremark $\displaystyle \V\left[ \sum_{k = 1}^{n} \cX_k \right] = \sum_{k = 1}^{n} \V[\cX]_k + 2 \sum_{k = 1}^{n - 1} \sum_{l = k + 1}^{n} \cov(\cX_k, \cX_l)$ + +\shortremark In Matrix-Notation für $\vec{\cX} = (\cX_1, \ldots, \cX_n)^\top$ +\[ + \Sigma = \begin{pmatrix} + \V[\cX_1] & \cov(\cX_1, \cX_2) & \dots & \cov(\cX_1, \cX_n) \\ + \cov(\cX_2, \cX_1) & \V[\cX_2] & \dots & \cov(\cX_2, \cX_n) \\ + \vdots & \vdots & \ddots & \ddots \\ + \cov(\cX_n, \cX_1) & \cov(\cX_n, \cX_2) & \dots & \V[\cX_n] \\ + \end{pmatrix} +\] diff --git a/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf b/semester4/ps/ps-jh/probability-and-statistics-cheatsheet.pdf index 35ba2656abdb448599acb17fd0fbf539e2b5bf6f..f0989c0a65ac16a49d66d4251a7b8c466133cf3a 100644 GIT binary patch delta 218283 zcmV(_K-9nH#uc5Y7O+nc0XUOT5N`rNHo=9=LI*ENR`hd2XF;-lYkK)lSPe>Y344NbQAERd3ucJP%kjQlZdTXuF4ObB-t2yLubJF`4FDyY zN>~C;a#4}Be)Re0Z}I&Lc$1u7UB(6lXmbNEGfsq%?qx=BL^Xf>{shL|uGS6! 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