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a/semester4/ps/ps-rb/parts/03_expectation.tex b/semester4/ps/ps-rb/parts/03_expectation.tex index 53dc602..14c2aef 100644 --- a/semester4/ps/ps-rb/parts/03_expectation.tex +++ b/semester4/ps/ps-rb/parts/03_expectation.tex @@ -130,7 +130,8 @@ $$ \begin{align*} \text{(i)}\quad & \text{cov}(X,Y) \geq 0 \\ \text{(ii)}\quad & \text{cov}(X,Y) = \text{cov}(Y,X) \\ - \text{(iii)}\quad & X,Y \text{ unabh. } \implies \text{cov}(X,Y) = 0\quad \text{\subtext{Nicht umgekehrt!}} \\ + \text{(iii)}\quad & X,Y \text{ unabh. } \implies \text{cov}(X,Y) = 0\\ + & \text{\color{gray}\footnotesize Nicht umgekehrt, $Y=X^2$ hat z.B. $\text{Cov}(X,Y)=0$} \\ \text{(iv)}\quad & \V[X \pm Y] = \V[X] + \V[Y] \pm 2\text{cov}(X,Y) \\ \text{(v)}\quad & \text{cov}\Biggl( \sum_{i=1}^{n}X_i,\sum_{j=1}^{n}Y_i \Biggr) = \sum_{i=1}^{n}\sum_{j=1}^{n}\text{cov}(X_i,Y_j) \\ \text{(vi)}\quad & \text{cov}(aX+b, cY+d) = ac\cdot\text{cov}(X,Y) \\ diff --git a/semester4/ps/ps-rb/parts/04_joint-distributions.tex b/semester4/ps/ps-rb/parts/04_joint-distributions.tex index c38b5ce..f4ea577 100644 --- a/semester4/ps/ps-rb/parts/04_joint-distributions.tex +++ b/semester4/ps/ps-rb/parts/04_joint-distributions.tex @@ -68,7 +68,8 @@ $$ % Bedingte Dichten fehlen noch, siehe HW5, F3 -\theorem \textbf{Erwartungswert} +\theorem \textbf{Erwartungswert}\\ +\subtext{z.B. $\E[XY] = \int_{-\infty}^{\infty}(xy)\cdot f_{X,Y}(x,y)$.} \begin{align*} &\E\Bigl[ \phi(X_1,\ldots,X_n) \Bigr] \\ &= \int_{-\infty}^\infty\cdots\int_{-\infty}^\infty\phi(x_1,\ldots,x_n)\cdot f(x_1\cdots x_n)\ dx_n\ldots dx_1 @@ -83,11 +84,25 @@ $$ {\scriptsize \remark Unabhängige stetige Variablen sind automatisch gemeinsam stetig + + \remark Für unabh. stetige Variablen muss der Träger ein karthesischs Produkt sein. z.B. für 2 Variabeln in $\R$ ginge $[0,1]\times[0,2]$ (Ein Rechteck) } % It's important to him to know \E[X]\E[Y] = \E[X\cdot Y] isn't enough for indep., requires \E[\phi(X)]\E[\psi(X)] instead \forall \phi,\psi -\notation \textbf{Bedingte gem. Verteilung} +\notation \textbf{Bedingte gem. Verteilung}\\ +\subtext{Analog zur Definition von $\P[X\sep Y]$.} $$ f_{X \vert Y} := \frac{f_{X,Y}(x,y)}{f_Y(y)} $$ + +{\footnotesize + \textbf{Beispiel}: Bei $f_{X,Y}=e^{-x},\ f_Y(y)=e^{-y}$ erhält man: + $$ + f_{X|Y} \overset{\text{def.}}{=} \frac{f_{X,Y}(x,y)}{f_Y(y)} = \frac{e^{-x}}{e^{-y}} = e^{-(x-y)} + $$ + Setzt man $z:=x-y$ gilt $f_{X|Y}\sim\text{Exp}(y)$. Gegeben $\{Y=y\}$ gilt: + $$ + X - y \sim \text{Exp}(1) \qquad \text{oder} \qquad X \sep Y=y \sim y + \text{Exp}(1) + $$ +} \ No newline at end of file diff --git a/semester4/ps/ps-rb/parts/06_limits.tex b/semester4/ps/ps-rb/parts/06_limits.tex index 1cacfcd..dbbec3a 100644 --- a/semester4/ps/ps-rb/parts/06_limits.tex +++ b/semester4/ps/ps-rb/parts/06_limits.tex @@ -50,7 +50,7 @@ $$ \definition \textbf{Standardisierung von $S_n$} $$ - S_n^* := \frac{S_n - n\mu}{\sigma\sqrt{n}} = \frac{S_n - \E[S_n]}{\sqrt{\V[S_n]}} + S_n^* := \frac{S_n - n\mu}{\sigma\sqrt{n}} = \frac{S_n - \E[S_n]}{\sqrt{\V[S_n]}} \sim \mathcal{N}(0,1) $$ \subtext{Im Skript auch $Z_n$} @@ -58,6 +58,26 @@ $$ \newpage +{\footnotesize + \textbf{Beispiel}: Approximation via Standardisierung mit ZGS. + + Sei $X_i,\ldots,X_n \overset{\text{i.i.d.}}{\sim} \P_X$ s.d. $\forall i: \E[X_i]=8, \V[X_i]=1.44$. + + für $S_{100} = \sum_{i=1}^{n}X_i$ approximieren wir $\P\Bigl[ 788 \leq S_100 \leq 824 \Bigr]$\\ + Es gilt: $\E[S_{100}] = 100\cdot\E[X_i] = 800,\quad \V[S_{100}]=100\cdot\V[X_i] = 144$ + \begin{align*} + &\P\Bigl[ 788 \leq S_100 \leq 824 \Bigr] \\ + = &\P\Biggl[\frac{788 - \E[S_{100}]}{\sqrt{\V[S_{100}]}} \leq \frac{S_{100} - \E[S_{100}]}{\sqrt{\V[S_{100}]}} \leq \frac{824 - \E[S_{100}]}{\sqrt{\V[S_{100}]}}\Biggr] \\ + = &\P\Bigl[ -1 \leq \frac{S_{100} - \E[S_{100}]}{\sqrt{\V[S_{100}]}} \leq 2 \Bigr] \\ + = & \Phi(2) - \Phi(-1) \\ + = & \Phi(2) - (1-\Phi(1)) \\ + = & \Phi(2) + \Phi(1) - 1 \\ + \approx & 0.8185 + \end{align*} + Genutztz wurde die Symmetrie von $\Phi$ und $\frac{S_{100} - \E[S_{100}]}{\sqrt{\V[S_{100}]}} \sim \mathcal{N}(0,1)$.\\ + \subtext{Gemäss ZGS Nur eine approximation, da wir hier $n=100$ nutzen.} +} + {\footnotesize \textbf{Beispiel}: Zentraler Grenzwertsatz als Teststatistik