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zI|0N*B?14vh!gr0pCXwjEj5gu(crJuMN2~+bcVjA)i8~?f3_I;D6-+EmOfp^dEw|SxP$%L$Bh48E^q-`*9DWA4vV1~2Y=u|w#C}(le0g~%7 z7pAFN!lrMnYf&KWF5+CbvstV8(&+9&_voITLgy%+r*3b1WiWaGIVtH^@t;xu%G(C+ X?FYXP^ZQ}ILQ>+y92~0pYQ+BnAvTnG diff --git a/wtheo9.tex b/wtheo9.tex index 60c051b..29cae1c 100644 --- a/wtheo9.tex +++ b/wtheo9.tex @@ -3,6 +3,7 @@ \title{Wtheo 0: Übungsblatt 9} \author{Josua Kugler, Christian Merten} \newcommand{\E}{\mathbb{E}} +\renewcommand{\P}{\mathbb{P}} \usepackage[]{mathrsfs} \newcommand{\cov}{\mathbb{C}\text{ov}} \newcommand{\var}{\mathbb{V}\text{ar}} @@ -81,7 +82,66 @@ \end{enumerate} \end{aufgabe} -\stepcounter{aufgabe} +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Es gilt + \begin{salign*} + \mathbb{F}^{Z_p}(x) &= \P^{Z_p}((-\infty, x])\\ + &= \P^{(-1)^{V_p}\cdot Y}((-\infty, x] \cap \{V_p = 0\}) + \P^{(-1)^{V_p}\cdot Y}((-\infty, x] \cap \{V_p = 1\})\\ + &= \P^Y((-\infty, x] \cap \{V_p = 0\}) + \P^{-Y}((-\infty, x] \cap \{V_p = 1\})\\ + &\stackrel{Y \indep V_p}{=} \P(\{Y \leq x\}) \cdot \P(\{V_p = 0\}) + \P(\{-Y \leq x\}) \cdot \P(\{V_p = 1\})\\ + &\stackrel{\text{Symmetrie } N_{(0,1)}}{=} \P(\{Y \leq x\}) \cdot \P(\{V_p = 0\}) + \P(\{Y \leq x\}) \cdot \P(\{V_p = 1\})\\ + &= \P(\{Y \leq x\}) \cdot \P(\{V_p = 0\}\cup \{V_p = 1\})\\ + &= \P(\{Y \leq x\})\\ + &= \mathbb{F}^{Y}(x)\\ + \end{salign*} + Daher gilt $Z_p \sim Y \sim N_{(0,1)}$. + \item Es gilt + \begin{salign*} + \P(\{Y < -1, Z_p < -1\}) &= \P(\{Y < -1\} \cap \{V_p = 0\}) + &\stackrel{Y \indep V_p}{=} \P(\{Y < -1\}) \cdot \P(\{V_p = 0\})\\ + &= (1-p) \cdot \P(\{Y < -1\}) + \end{salign*} und völlig analog + \begin{salign*} + \P(\{Y < -1, Z_p > 1\}) &= \P(\{Y < -1\} \cap \{V_p = 1\}) + &\stackrel{Y \indep V_p}{=} \P(\{Y < -1\}) \cdot \P(\{V_p = 1\})\\ + &= p \cdot \P(\{Y < -1\}) + \end{salign*} + Angenommen, $Y \indep Z_p$. Dann gilt + \begin{salign*} + \P(\{Y < -1, Z_p < -1\}) &= \P(\{Y< -1\})\P(\{Z_p < -1\})\\ + (1-p) \cdot \P(\{Y < -1\}) &\stackrel{\text{(a)}}{=} \P(\{Y < -1\})^2\\ + (1-p) &= \P(\{Y < -1\}) + \end{salign*} und völlig analog + \begin{salign*} + \P(\{Y < -1, Z_p > 1\}) &= \P(\{Y< -1\})\P(\{Z_p > 1\})\\ + p \cdot \P(\{Y < -1\}) &\stackrel{\text{(a), Symmetrie } N_{(0,1)}}{=} \P(\{Y < -1\})^2\\ + p &= \P(\{Y < -1\}) + \end{salign*} + Nun führen wir eine Fallunterscheidung durch. + Für $p = \frac{1}{2}$ folgt $\P(\{Y < -1\}) < \P(\{Y < 0\}) \leq \frac{1}{2}$, Widerspruch zu $p = \P(\{Y < -1\})$. + Für $p \neq \frac{1}{2}$ erhalten wir aus $p = \P(\{Y < -1\}) = (1-p)$ ebenfalls einen Widerspruch. + Daher ist $Y \not \indep Z_p$. + \item Es gilt + \begin{salign*} + \E(YZ_p) &= \int_\R \int_\R yz \mathbbm{f}^{Y, Z_p}(y, z) \d{y}\d{z}\\ + &\stackrel{Z_p = (-1)^{V_p}Y}{=} \int_{0,1} \int_\R y^2 (-1)^v \mathbbm{f}^{Y, V_p}(y, v) \d{y}\d{v}\\ + &\stackrel{Y \indep V_p}{=} (1-p) \cdot \int_\R y^2 \mathbbm{f}^Y(y) \d{y} + p\cdot \int_\R -y^2 \mathbbm{f}^Y(y) \d{y}\\ + &= (1 - 2p) \int_\R y^2 \mathbbm{f}^Y(y) \d{y} + \end{salign*} + Außerdem gilt + \begin{salign*} + \E(y)\E(Z_p) &= \int_\R y\mathbbm{f}^Y(y) \d{y} \int_\R z\mathbbm{f}^{Z_p}(z) \d{z}\\ + &\stackrel{Z_p = (-1)^{V_p}Y}{=} \int_\R y\mathbbm{f}^Y(y) \d{y} \cdot \int_{0,1} \int_\R y (-1)^v \mathbbm{f}^{Y, V_p}(y, v) \d{y}\d{v}\\ + &\stackrel{Y \indep V_p}{=} (1-p) \cdot \left(\int_\R y \mathbbm{f}^Y(y) \d{y}\right)^2 - p\cdot \left(\int_\R y^2 \mathbbm{f}^Y(y) \d{y}\right)^2\\ + &= (1 - 2p) \left(\int_\R y \mathbbm{f}^Y(y) \d{y}\right)^2 + \end{salign*} + Für $p = 0$ erhalten wir daher + \begin{align*} + \cov(Y, Z_p) = \E(YZ_p) - \E(Y)\E(Z_p) = 0 - 0 = 0. + \end{align*} + \end{enumerate} +\end{aufgabe} \begin{aufgabe} \begin{enumerate}[(a)]