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Übungsblatt 10} +\author{Josua Kugler, Christian Merten} +\usepackage[]{mathrsfs} +\newcommand{\E}{\mathbb{E}} + +\begin{document} + +\punkte[36] + +\begin{aufgabe} + Sei $X \in \mathscr{A}^{n}$ die Flughöhe von $n$ Barock-Raketen und $Y \in \mathscr{A}^{m}$ die Flughöhe + von $m$ Renaissance-Raketen. Laut + Aufgabenstellung ist $(X,Y) \sim (N_{(\mu_B, \sigma^2)}^{n} \otimes N_{(\mu_R, \sigma^2)}^{m})$. + Sei außerdem $\mathscr{H}_0\colon \mu_B \ge \mu_R$. Nach Satz 26.43 hält dann + der linksseitige Test + \[ + \varphi_c^{l} = \mathbbm{1}_{ \{ \overline{X}_n - \overline{Y}_{m} \le -c \frac{\sqrt{n + m} }{\sqrt{nm} } \hat{S}_{n,m}\}} = \mathbbm{1}_{\left\{ - \frac{\overline{X}_n - \overline{X}_m}{\hat{S}_{n,m}} \frac{\sqrt{nm} }{\sqrt{n+m} } \ge c\right\} } + \] mit $c = t_{(n+m-2),(1-\alpha)}$ das Niveau $\alpha$ ein. + Einsetzen aller Werte ergibt + \[ + - \frac{\overline{X}_n - \overline{X}_m}{\hat{S}_{n,m}} \frac{\sqrt{nm} }{\sqrt{n+m} } + \approx 0.941 < 1.734 = t_{18,0.95} + .\] Also kann $\mathscr{H}_0$ nicht zum Signifikanzniveau $0.05$ abgelehnt werden. +\end{aufgabe} + +\stepcounter{aufgabe} + +\begin{aufgabe} + Sei $X, X_n\colon \Omega \to \R$ für $n \in \N$. + $X_n \xrightarrow{\mathbb{P}\text{ f.s.}} X \implies X_n \xrightarrow{\mathbb{P}} X$ nach VL. + Sei also $X_n \xrightarrow{\mathbb{P}} X$. Sei weiter + $\mathcal{X} \coloneqq \{ \omega \in \Omega \mid \mathbb{P}(\omega) > 0\} $. Dann + ist $\mathbb{P}(\Omega \setminus \mathcal{X}) = 0$. Es genügt also + zu zeigen, dass $\lim_{n \to \infty} |X_n(\omega) - X(\omega)| = 0$ für $\omega \in \mathcal{X}$. + Sei dazu $\epsilon > 0$ und $\omega \in \mathcal{X}$. + Da $\lim_{n \to \infty} \mathbb{P}(|X_n - X| > \epsilon) = 0$ ex. + ein $n_0 \in \N$, s.d. $\forall n \ge n_0$: + \[ + \mathbb{P}(|X_n - X| > \epsilon) < \mathbb{P}(\omega) + .\] Damit folgt $w \not\in \{|X_n - X| > \epsilon \} $, also + $|X_n(\omega) - X(\omega)| = 0$. +\end{aufgabe} + +\begin{aufgabe} + Zunächst berechne für $n \in \N$: + \begin{salign*} + \mathbb{P}^{U}([n, \infty)) &= 1 - \mathbb{P}^{U}((-\infty, n)) + = 1 - \int_{0}^{n} \exp(-v) \d{v} = \exp(-n) \\ + \mathbb{P}^{V}([n, \infty)) &= 1 - \mathbb{P}^{V}((-\infty, n)) + = 1 - \int_{1}^{n} \frac{1}{v^2} \d{v} = \frac{1}{n} + .\end{salign*} + \begin{enumerate}[(a)] + \item Sei $\epsilon > 0$ und $n \in \N$ mit $n > \epsilon$. Dann gilt + \begin{salign*} + \mathbb{P}(|X_n| > \epsilon) = \mathbb{P}(n \mathbbm{1}_{[n, \infty)}(U) > \epsilon) \; + \stackrel{n > \epsilon}{=} \; \mathbb{P}(\mathbbm{1}_{[n, \infty)}(U) > 0 ) + = \mathbb{P}^{U}([n, \infty)) = \exp(-n) + .\end{salign*} + Also folgt $\lim_{n \to \infty} \mathbb{P}(|X_n| > \epsilon) = 0$ also + $X_n \xrightarrow{\mathbb{P}} 0$. + + Sei nun $\sqrt{n} > \epsilon$. Dann gilt + \begin{salign*} + \mathbb{P}(|Y_n| > \epsilon) = \mathbb{P}(\sqrt{n} \mathbbm{1}_{[n, \infty)}(V) > \epsilon) + \; \stackrel{\sqrt{n} > \epsilon}{=} + \mathbb{P}^{V}([n, \infty)) = \frac{1}{n} + .\end{salign*} + Also folgt $\lim_{n \to \infty} \mathbb{P}(|Y_n| > \epsilon) = 0$ also + $Y_n \xrightarrow{\mathbb{P}} 0$. + \item Betrachte + \begin{salign*} + \E(|X_n|^2) = \E(n^2 \mathbbm{1}_{[n, \infty)}(U)) + = n^2 \int_{\R}^{} \mathbbm{1}_{[n, \infty)}(v) f^{U}(v) \d{v} + = n^2 \mathbb{P}^{U}((n, \infty)) + = n^2 \exp(-n) + .\end{salign*} + Betrachte $f(x) \coloneqq x^2 \exp(-x) \in C^{\infty}(\R)$. + Dann ist durch mehrfache Anwendung von de l'Hospital + ($*$): + \[ + \lim_{x \to \infty} f(x) = \lim_{x \to \infty} x^2 \exp(-x) = \lim_{x \to \infty} \frac{x^2}{\frac{1}{\exp(-x)}} + \stackrel{(*)}{=} \lim_{x \to \infty} \frac{2x}{\exp(x)} + \stackrel{(*)}{=} \lim_{x \to \infty} \frac{2}{\exp(x)} = 0 + .\] + Mit der Folge $(a_n)_{n \in \N}$ mit $a_n \coloneqq n$ folgt also + $\lim_{n \to \infty} n^2\exp(-n) = f(n) = \lim_{x \to \infty} f(x) = 0$. + Also folgt insgesamt $\lim_{n \to \infty} \Vert X_n \Vert_{L^2} = 0$ und damit + $X_n \xrightarrow{\mathscr{L}_2} 0$. + + Weiter folgt + \begin{salign*} + \E(|Y_n|^2) = \E(n\mathbbm{1}_{[n, \infty)}(V)) + = n \int_{\R}^{} \mathbbm{1}_{[n, \infty)}(v) f^{V}(v) \d{v} + = n \mathbb{P}^{V}((n, \infty)) + = n \frac{1}{n} = 1 + .\end{salign*} + Damit folgt $\lim_{n \to \infty} \Vert Y_n \Vert_{L^2} = \sqrt{1} = 1 \neq 0$. Da + $Y_n \xrightarrow{\mathbb{P}} 0$ konvergiert $Y_n$ nicht in $\mathscr{L}_2$ gegen + ein $Y \in \overline{\mathscr{A}}$ mit $Y \neq 0$ $\mathbb{P}$ f.s., da + sonst auch $Y_n \xrightarrow{\mathbb{P}} Y \neq 0$ und + stochastische Grenzwerte $\mathbb{P}$ f.s. übereinstimmen. + Also konvergiert $Y_n$ nicht in $\mathscr{L}_2$. + \end{enumerate} +\end{aufgabe} + +\end{document}