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z37NP!2pI{Lp&6v?%^iL$&i}DQ33YhcSQr^uImMXRxkT6*IfYn-*;&Lzg;|BfggDtb z#8{d53IG2oa6d1SHM6&HwIpO?WlAex2BQLyEy>u*8lm>z-E8r+vDjJh{63ShYAMA- z6;oQOv}Ss$9Bam>o0N7nBN|v7G+7=Vgr=E*;)#T7(6AY9Bt^!dK|&6C5Mb~7c-ZE7 z@H;)(K@|Rn6Pi#Rj=u&ZbTD@&$cDWk4<$N3tc`;v0E_n$j!rK8Nt|w@~=T`8bLk|$x{vj z!4A@625mFK`WS+V8!=o4ecE?5f{yTTmJ$lEeLqTTKeWI38rS+DRbUn*hEfxhJ06cSk z|J}x^Q$hZu_R2|Z_gTA$87_M1E+x&)c*#utcT&Bo6`A+Z=PiG*nyi0H<_{RM5kCm+ zdQSAs0:\qquad \P[X \geq c] \leq \frac{\E[g(X)]}{g(c)} +$$ + +\theorem \textbf{Jensen}\\ +\smalltext{$\phi:\R\to\R$ konvex,$\quad \E[\phi(X)],\E[X]$ wohldefiniert} +$$ + \phi\Bigl(\E[X]\Bigr)\leq\E\Bigl[\phi(X)\Bigr] +$$ + +\lemma \textbf{Dreiecksungleichung}\\ +\subtext{Jensen mit $\phi(X) = |X|$ und $\phi(X) = X^2$} +$$ + \Bigl\vert\E[X]\Bigr\vert\leq\E\Bigl[\vert X\vert\Bigr] \qquad \E[|X|] \leq \sqrt{\E[X^2]} +$$ + +\theorem \textbf{Chebychev}\\ +\subtext{$Y$ s.d. $\V[Y] < \infty,\quad c>0$} +$$ + \P\Bigl[ |Y-\E[Y]| \geq c \Bigr] \leq \frac{\V[Y]}{c^2} +$$ + +\newpage +\subsection{Varianz} + +\definition \textbf{Varianz}\\ +\subtext{$\E[X^2]<\infty$} +$$ + \mathbb{V}[X] := \E\Bigl[ (X - \E[X])^2 \Bigr] +$$ +\definition \textbf{Standardabweichung} +$$ + \rho(X) := \sqrt{\mathbb{V}[X]} +$$ +{\scriptsize + \notation Auch $\rho, \rho_X$ +} + +\lemma \textbf{Varianz} (Alternativ) +$$ + \mathbb{V}[X] = \E[X^2]-\E[X]^2 +$$ + +\lemma \textbf{Eigenschaften} +\begin{align*} + \text{(i)} &\quad \V[X] \geq 0 \\ + \text{(ii)} &\quad \V[aX] = a^2\V[X] \\ + \text{(iii)} &\quad \V[X+a] = \V[X] +\end{align*} + +\lemma \textbf{Addition} (Unabhängigkeit)\\ +\smalltext{$X_1,\ldots,X_n$ paarweise unabhängig} +$$ + \V\Biggl[ \sum_{k=1}^{n}X_k \Biggr] = \sum_{k=1}^{n}\V[X_k] +$$ + +\newpage +\subsection{Kovarianz} + +\definition \textbf{Kovarianz}\\ +\subtext{$X,Y$ s.d. $\E[X^2],\E[Y^2]<\infty$} +$$ + \text{cov}(X,Y) := \E\Bigl[ (X-\E[X])\cdot(Y-\E[Y]) \Bigr] +$$ + +\lemma \textbf{Kovarianz} (Alternativ) +$$ + \text{cov}(X,Y) = \E[XY] - \E[X]\E[Y] +$$ + +\remark $\text{cov}(X,X) = \V[X]$ + +\lemma $X,Y$ unabh. $\implies \text{cov}(X,Y)=0$\\ +\subtext{Nicht umgekehrt gültig} +% Gegenbeispiel: Slides p.240 + +\lemma \textbf{Eigenschaften von} $\text{cov}$ + +\definition \textbf{Kovarianzmatrix} +$$ + \Sigma = \text{cov}(\textbf{X}) = \begin{bmatrix} + \V[X_1] & \text{cov}(X_1,X_2) & \cdots & \text{cov}(X_1,X_n) \\ + \text{cov}(X_2,X_1) & \text{cov}(X_2,X_2) & \cdots & \text{cov}(X_2,X_n) \\ + \vdots & \vdots & \ddots & \vdots \\ + \text{cov}(X_n,X_1) & \text{cov}(X_n,X_2) & \cdots & \text{cov}(X_n,X_n) + \end{bmatrix} +$$ +\subtext{$\textbf{X} = (X_1,\ldots,X_n)^\top$} \ No newline at end of file diff --git a/semester4/ps/ps-rb/util/helpers.tex b/semester4/ps/ps-rb/util/helpers.tex index 4f998c9..0094575 100644 --- a/semester4/ps/ps-rb/util/helpers.tex +++ b/semester4/ps/ps-rb/util/helpers.tex @@ -60,6 +60,8 @@ \def \P{\mathbb{P}} \def \F{\mathcal{F}} \def \E{\mathbb{E}} +\def \I{\mathbb{I}} +\def \V{\mathbb{V}} % Titles \def \definition{\colorbox{lightgray}{Def} }