[PS] Variance and covariance complete

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: $\P[|\cY - \E[\cY]| \geq c] \leq \frac{\V[\cY]}{c^2}$

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 $\vec{\cX} = (\cX_1, \ldots, \cX_n)^\top$
+\[
+    \Sigma = \begin{pmatrix}
+        \V[\cX_1]          & \cov(\cX_1, \cX_2) & \dots  & \cov(\cX_1, \cX_n) \\
+        \cov(\cX_2, \cX_1) & \V[\cX_2]          & \dots  & \cov(\cX_2, \cX_n) \\
+        \vdots             & \vdots             & \ddots & \ddots             \\
+        \cov(\cX_n, \cX_1) & \cov(\cX_n, \cX_2) & \dots  & \V[\cX_n]          \\
+    \end{pmatrix}
+\]

semester4/ps/ps-jh/probability-and-statistics-cheatsheet.tex

@@ -11,11 +11,15 @@
 \loadGerman
 \setnumberingpreset{off}
 \renewcommand{\definitionShortNamingDE}{Def}
+\renewcommand{\propositionShortNamingDE}{Prop}
 \renewcommand{\remarkShortNamingDE}{Bem}
 
 \setsubsectionnumbering{section}
 \renewcommand{\examplenumbering}{off}
 
+\newcommand{\cov}{\text{cov}}
+\renewcommand{\vec}[1]{\bm{#1}}
+
 \begin{document}
 \startDocument
 \noverticalspacing
@@ -67,6 +71,7 @@
 \input{parts/03_expected-value/03_properties.tex}
 \input{parts/03_expected-value/04_inequalities.tex}
 \input{parts/03_expected-value/05_variance.tex}
+\input{parts/03_expected-value/06_covariance.tex}
 % \input{parts/03_expected-value/}