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\mathbbm{1}_{\{|x| \le 1\}} \int_{-\sqrt{1-x^2} }^{\sqrt{1-x^2} } \frac{1}{\pi} \d{y} \\ + &= \frac{2}{\pi} \sqrt{1-x^2} \mathbbm{1}_{\{|x| \le 1\}} + \intertext{ + Ganz analog folgt für $y \in \R$:} + f^{Y}(y) &= \frac{2}{\pi} \sqrt{1-y^2} \mathbbm{1}_{\{|y| \le 1\} } + .\end{salign*} + \item Anwenden der Formel der VL ergibt zunächst mit Anwendung des Transformationssatzes + \begin{salign*} + \E(X) &= \int_{\R}^{} x f^{X}(x) \d{x} \\ + &= \frac{1}{\pi} \int_{-1}^{1} 2x \sqrt{1-x^2} \d{x} \\ + &= \frac{1}{\pi} \left[ \int_{-1}^{0} 2x \sqrt{1-x^2} \d{x} + + \int_{0}^{1} 2x \sqrt{1-x^2} \d{x} \right] \\ + &\stackrel{z = 1-x^2}{=} \frac{1}{\pi} \left[ \int_{0}^{1} - \sqrt{z} \d{z} + + \int_{1}^{0} - \sqrt{z} \d{z} \right] \\ + &= 0 + \intertext{Unter erneuter Formelanwendung folgt} + \E(X^2) &= \int_{\R}^{} x^2 f^{X}(x) \d{x} \\ + &= \frac{2}{\pi} \int_{-1}^{1} x^2 \sqrt{1-x^2} \d{x} \\ + &\stackrel{x = \sin(\varphi)}{=} \frac{2}{\pi} \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} + \sin^2(\varphi) \cos^2(\varphi)\d{\varphi} \\ + &= \frac{2}{\pi} \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{1}{4} \sin^2(2\varphi) \d{\varphi} \\ + &\stackrel{\psi = 2 \varphi}{=} + \frac{1}{2\pi} \int_{-\pi}^{\pi} \sin^2(\psi) \frac{1}{2} \d{\psi} \\ + &= \frac{1}{4 \pi} \int_{-\pi}^{\pi} \frac{1}{2}(\sin^2(\psi) + \cos^2(\psi)) \d{\psi} \\ + &= \frac{1}{8 \pi} 2 \pi \\ + &= \frac{1}{4} + \intertext{Damit folgt} + \var(X) &= \frac{1}{4} + \intertext{Ganz analog} + \var(Y) &= \frac{1}{4} + \intertext{Betrachte nun zunächst} + \int_{0}^{2\pi} \sin(\varphi) \cos(\varphi)\d{\varphi} + &\stackrel{\text{part. Integrat.}}{=} \underbrace{\sin^2\varphi \Big|_{0}^{2\pi}}_{= 0} + - \int_{0}^{2\pi} \sin\varphi \cos\varphi \d{\varphi} \\ + \intertext{Damit folgt} + \int_{0}^{2\pi} \sin\varphi \cos\varphi \d{\varphi} &= 0 + \tageq \label{eq:1} + \intertext{Es ist $\int_{\R^2}^{} \left|\frac{xy}{\pi} \right| \mathbbm{1}_{\{(x,y) \in E\}} + \d{(x,y)} \le \frac{1}{\pi}\mathscr{L}^{2}(E) = 1 < \infty$, + d.h. Fubini ist anwendbar. Damit folgt} + \E(XY) &= \int_{\R^2}^{} xy f^{X,Y}(x,y) \d{(x,y)} \\ + &= \int_{\R}^{} \int_{\R}^{} \frac{xy}{\pi} \mathbbm{1}_{\{(x,y) \in E\} } \d{x} \d{y} \\ + &\stackrel{\text{Trafosatz}}{=} + \frac{1}{\pi}\int_{0}^{1} \d{r} \int_{0}^{2\pi} \d{\varphi} r^{3} \cos(\varphi) \sin(\varphi) \\ + &= \frac{1}{4\pi} \int_{0}^{2\pi} \cos(\varphi) \sin(\varphi)\d{\varphi} \\ + &\stackrel{\text{(\ref{eq:1})}}{=} 0 + \intertext{Da $\E(X) = \E(Y) = 0$ folgt also} + \cov(X,Y) &= 0 + \intertext{Und damit} + \rho(X,Y) &= 0 + .\end{salign*} + \item Es gilt nach VL: $X \indep Y \iff f^{X,Y}(x,y) = f^{X}(x) f^{Y}(y)$ $\mathscr{L}$-f.ü. Nun + betrachte $A \coloneqq (-1,1)^2 \setminus E$. Dann ist + \[ + \mathscr{L}^2(A) = \mathscr{L}^2(A) - \mathscr{L}^2(E) = 2^2 - \pi = 4 - \pi > 0 + .\] Also ist $A$ keine $\mathscr{L}$-Nullmenge. Jedoch gilt $\forall (x,y) \in A$: + \[ + f^{X,Y}(x,y) = 0 \neq \frac{4}{\pi^2} \underbrace{\sqrt{1-x^2}}_{> 0} \underbrace{\sqrt{1-y^2} }_{> 0} + .\] Also folgt $X$ und $Y$ nicht unabhängig. + \end{enumerate} +\end{aufgabe} + +\stepcounter{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Rechnen ergibt für $z \in \R$ + \begin{salign*} + \mathbb{F}^{M_1}(z) &= \mathbb{P}\left( \{M_1 \le z\} \right) \\ + &= \mathbb{P}\left( \left\{ \min_{i \in \{1, \ldots, n\} }{X_i} \le z\right\} \right) \\ + &= 1 - \mathbb{P}\left( \bigcap_{i=1}^{n} \{X_i > z\} \right) \\ + &\stackrel{\text{unabh.}}{=} + 1 - \prod_{i=1}^{n} \mathbb{P}(\{X_i > z\}) \\ + &\stackrel{\text{idv}}{=} 1 - (1 - \mathbb{F}^{X}(z))^{n} \\ + \mathbb{F}^{M_2}(z) &= \mathbb{P}\left( \{M_2 \le z\} \right) \\ + &= \mathbb{P}\left( \prod_{i=1}^{n} \{X_i \le z\} \right) \\ + &\stackrel{\text{unabh.}}{=} \prod_{i=1}^{n} \mathbb{P}(\{X_i \le z\}) \\ + &\stackrel{\text{idv}}{=} \mathbb{F}^{X}(z)^{n} + .\end{salign*} + \item Für $X_1 \sim \text{Exp}_{\lambda}$ ist $\mathbb{F}^{X_1} = (1 - \exp(- \lambda z))\mathbbm{1}_{\R^{+}}$. Damit folgt + \begin{salign*} + \mathbb{F}^{M_1}(z) &= 1 - (1 - \mathbb{F}_{\text{Exp}_{\lambda}}(z))^{n} \\ + &= 1 - \left[ 1 - \left( 1 - \exp(-\lambda z) \right) \mathbbm{1}_{\R^{+}(z)} \right]^{n} \\ + &= 1 - \left[ 1 - \mathbbm{1}_{\R^{+}}(z) + \exp(- \lambda z) \mathbbm{1}_{\R^{+}}(z)\right]^{n}\\ + &= 1 - \begin{cases} + \exp(- \lambda n z) & z \in \R^{+} \\ + 1 & z \not\in \R^{+} + \end{cases}\\ + &= (1 - \exp(- \lambda n z)) \mathbbm{1}_{\R^{+}} \\ + &= \mathbb{F}_{\text{Exp}_{n\lambda}} + .\end{salign*} + Da die Verteilungsfunktion das W-Maß eindeutig festlegt, folgt + $M_1 \sim \text{Exp}_{n\lambda}$. + \item Für $X_1 \sim U_{[0, \theta]}$ ist die Dichte $f(x) = \frac{1}{\theta} \mathbbm{1}_{[0, \theta]}$ + gegeben. Damit folgt + \begin{salign*} + \E_{\theta}(X_1) &= \int_{\R}^{} \frac{x}{\theta} \mathbbm{1}_{[0, \theta]} \d{x} \\ + &= \frac{1}{\theta} \int_{0}^{\theta} x \d{x} \\ + &= \frac{\theta}{2} \\ + \E_{\theta}(X_1^2) &= \int_{\R}^{} \frac{x^2}{\theta} \mathbbm{1}_{[0, \theta]} \d{x} \\ + &= \frac{1}{\theta} \int_{0}^{\theta} x^2 \d{x} \\ + &= \frac{\theta^2}{3} \\ + \var_{\theta}(X_1) &= \E_{\theta}(X_1^2) - \E_{\theta}(X_1)^2 \\ + &= \frac{\theta^2}{3} - \frac{\theta^2}{4} \\ + &= \frac{\theta^2}{12} + .\end{salign*} + Es gilt $f^{M_{2}} = (\mathbb{F}^{M_{2}})'$. Damit folgt für $z \in \R$: + \begin{salign*} + f^{M_2}(z) &= (\mathbb{F}^{M_2})'(z) \\ + &= (\mathbb{F}^{X}(z)^{n})' \\ + &= n (\mathbb{F}^{X}(z)^{n-1}) f^{X}(z) \\ + &= n \left( \frac{(z \land \theta) \lor \theta}{\theta} \right)^{n-1} + \frac{1}{\theta} \mathbbm{1}_{[0, \theta]}(z) \\ + &= \frac{n}{\theta^{n}} z^{n-1} \mathbbm{1}_{[0, \theta]}(z) + \intertext{Damit folgt} + \E_{\theta}(M_2) &= \int_{\R}^{} x \frac{n x^{n-1}}{\theta ^{n}} \mathbbm{1}_{[0, \theta]} + \d{x} \\ + &= \frac{n}{\theta^{n}} \int_{0}^{\theta} x^{n} \d{x} \\ + &= \frac{n}{n+1} \theta + .\end{salign*} + \item Rechnen ergibt + \begin{salign*} + \E_{\theta}(\overline{X}_n) &= \E_{\theta}\left( \frac{1}{n} \sum_{k=1}^{n} X_k \right) \\ + &\stackrel{\text{lin.}}{=} \frac{1}{n} \sum_{k=1}^{n} \E_{\theta}(X_k) \\ + &\stackrel{\text{idv}}{=} \E_{\theta}(X_1) \\ + &= \frac{\theta}{2} + \intertext{Damit folgt} + \text{Bias}_{\theta}(\hat{\theta}_1) &= \E_{\theta}(\hat{\theta}_1 - \theta) \\ + &= 2 \E_{\theta}(\overline{X}_n) - \theta \\ + &= 2 \frac{\theta}{2} - \theta \\ + &= 0 \\ + \text{Bias}_{\theta}(\hat{\theta}_2) + &= \E_{\theta}(\hat{\theta}_2 - \theta) \\ + &= \E_{\theta}(M_2) - \theta \\ + &= \frac{n}{n+1} \theta - \theta \\ + &= - \frac{\theta}{n+1} + \intertext{Nun rechne} + \E_{\theta}(M_2^2) &= \int_{\R}^{} x^2 \frac{n x^{n-1}}{\theta^{n}} \mathbbm{1}_{[0, \theta]}(x) \d{x} \\ + &= \frac{n}{\theta ^{n}} \int_{0}^{\theta} x^{n+1} \d{x} \\ + &= \frac{n}{n+2} \theta^2 \tageq \label{eq:2} + \intertext{ + Es ist offensichtlich $\hat{\theta}_3$ nun erwartungstreu. Damit folgt für $n > 1$} + \var_{\theta}(\hat{\theta}_1) &= 4 \var_{\theta}(\overline{X}_n) \\ + &\stackrel{\text{unabh.}}{=} \frac{4}{n^2} \sum_{k=1}^{n} \var_{\theta}(X_k) \\ + &\stackrel{\text{idv}}{=} \frac{4}{n} \var_{\theta}(X_1) \\ + &= \frac{\theta^2}{3n} \\ + &> \frac{\theta^2}{n(n+2)} \\ + &= \left( \frac{n+1}{n} \right)^2 \frac{n}{n+2} \theta^2 - \theta^2 \\ + &\stackrel{\text{(\ref{eq:2})}}{=} \left( \frac{n+1}{n} \right)^2 \E_{\theta}(M_2^2) - \E_{\theta}\left( \frac{n+1}{n} M_2 \right)^2 \\ + &= \E_{\theta} (\hat{\theta}_3^2) - \E_{\theta}(\hat{\theta}_3)^2 \\ + &= \var_{\theta}(\hat{\theta}_3) + \intertext{Schlussendlich ergibt sich} + \text{MSE}_{\theta}(\hat{\theta}_1) &= \E_{\theta}(|\hat{\theta}_1 - \theta|^2) \\ + &= 4 \E_{\theta}(\overline{X}_n^2) - \theta^2 \\ + &= 4 \var_{\theta}(\overline{X}_n) + 4 \E_{\theta}(\overline{X}_n)^2 - \theta^2 \\ + &\stackrel{\text{iid}}{=} \frac{4}{n} \var_{\theta}(X_1) \\ + &= \frac{\theta^2}{3n} \\ + \text{MSE}_{\theta}(\hat{\theta}_2) &= \E_{\theta}(\hat{\theta}_2^2) - 2 \theta \E_{\theta}(M_2) + + \theta^2 \\ + &= \frac{n}{n+2} \theta^2 - 2 \frac{n}{n+1} \theta^2 + + \theta^2 \\ + &= \frac{2 \theta^2}{(n+2)(n+1)} \\ + \text{MSE}_{\theta}(\hat{\theta}_3) &= \E_{\theta}(\hat{\theta}_3^2) - 2 \theta \E_{\theta} + (\hat{\theta}_3) + \theta^2 \\ + &= \left( \frac{n+1}{n} \right)^2 \frac{n}{n+2} \theta^2 - \theta^2 \\ + &= \frac{\theta^2}{n(n+2)} + .\end{salign*} + \end{enumerate} +\end{aufgabe} + +\end{document}