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z#~b2RP2h{ny#bnl(2xA8w+HX1H(Ur;uRWIgDa0=nhEdrVkDhw%TO=NNQ2O0%QtrXG z-XL(9NB=UnCOjwhGC&g5oVcr6QC){onWo8CMG;QY9-1^k68}+Tl`1V^?Y^q)xaMt{ zT;|e?Y6=?-go?WFZ`g9%H`;|iEvJ{+r37sU{4uMl#Q%3rIyoCSI=ef5Cn4i^ZZWgN Ll9Gzbi^2Xk;An1& diff --git a/wtheo3.tex b/wtheo3.tex index b54c840..f383b5a 100644 --- a/wtheo3.tex +++ b/wtheo3.tex @@ -1,4 +1,4 @@ -\documentclass[uebung]{../../../lecture} +\documentclass[uebung]{lecture} \title{Wtheo 0: Übungsblatt 3} \author{Josua Kugler, Christian Merten} @@ -54,7 +54,28 @@ \end{enumerate} \end{aufgabe} -\stepcounter{aufgabe} +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Ein Neyman-Pearson-Test für dieses Testproblem ist gegeben durch die Funktion $\mathbbm{1}_{A_k}$ mit + \begin{align*} + A_k &= \{x \colon \mathbbm{p}_{\mathrm{Poi}_{\lambda_1}} (x) \geq \mathbbm{p}_{\mathrm{Poi}_{\lambda_0}} (x)\}\\ + &= \{x \colon e^{-\lambda_1} \frac{\lambda_1^x}{x!} \geq k e^{-\lambda_0} \frac{\lambda_0^x}{x!}\}\\ + &= \{x \colon e^{\lambda_0 -\lambda_1} \frac{\lambda_1^x}{\lambda_0^x} \geq k\} + \end{align*} + Damit einer dieser Tests ein bester Test zum Niveau $\alpha \in (0,1)$ ist, muss $\mathbbm{P}_{\lambda_0}(A_k) = \alpha$ gelten. + \item Da $e^{\lambda_0 -\lambda_1} \frac{\lambda_1^x}{\lambda_0^x}$ für $\lambda_1 > \lambda_0$ und $x > 0$ + stets streng monoton wachsend ist und $\mathbbm{P}_{\lambda_0}(A_k) = \alpha$ völlig unabhängig von $\lambda_1$ ist, + muss jeder beste Test zum Niveau $\alpha$ auch ein gleichmäßig bester Test für $H_0$ gegen $H_1'$ sein. + \item Wählen wir $A = [9157, \infty)$ als Ablehnungsbereich, so erhalten wir + \[ + \mathbbm P_0 (A) = \sum_{k = 9157}^{\infty} \frac{\lambda_0^k}{k!} e^{-\lambda_0} \leq 0.05. + \] + Für ein beliebiges $\lambda_1$ existiert jetzt ein $k$ derart, dass wir diesen Ablehnungsbereich als Neyman-Pearson-Test schreiben können. + \[ + \{x \colon e^{\lambda_0 -\lambda_1} \frac{\lambda_1^x}{\lambda_0^x} \geq k\} = \{9157,\dots\}. + \] + \end{enumerate} +\end{aufgabe} \begin{aufgabe} Zunächst ist zu bemerken, dass mit $\mathscr{E} := \{ (a, \infty] \mid a \in \R\} $ nach VL @@ -157,4 +178,37 @@ \end{enumerate} \end{aufgabe} +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Sei $[a,b]$ ein Intervall in $\mathscr B(\R)$. + Sei dann $A \coloneqq f^{-1}([a,n])$ und $\alpha \in U_\epsilon(\inf A) \cap A$ sowie $\beta \in U_\epsilon(\sup A) \cap A$. + Für beliebiges $\alpha \le x \le \beta$ folgt aufgrund der Monotonie $a \leq f(\alpha) \le f(x) \le f(\beta) \le b$, + also $f(x) \in [a,b]$ und damit $x\in A$. + Also muss $A$ ein Intervall sein und damit wieder in $\mathscr B(\R)$ liegen. + Da die Menge aller Intervalle $[a,b]$ bereits ein Erzeuger von $\mathscr B(\R)$ ist, folgt daraus bereits die Messbarkeit. + \item Sei $(s_n)_{n\in \N}$ eine reelle Folge mit $\lim\limits_{n \to \infty} s_n = s$. + Dann ist $(g_n)_{n\in\N},\; g_n(x) \coloneqq g(s_n, x)$ eine Funktionenfolge, die wegen + \[ + \lim\limits_{n \to \infty} \lVert g(s_n,x) - g(s,x)\rVert = \lVert \lim\limits_{n \to \infty} g(s_n,x) - g(s,x)\rVert = 0 + \] + gleichmäßig konvergiert. Insbesondere gilt also + \[ + \lim\limits_{n \to \infty} h(s_n) = \lim\limits_{n \to \infty} \int_0^1 g(s_n,x) \d{x} + = \int_0^1 \lim\limits_{n \to \infty} g(s_n,x) \d{x} = \int_0^1 g(s,x) = h(s). + \] + \item Wähle ein $A$, sodass $[0,1] \subset A \in 2^\R$, aber $A \notin \mathscr B(\R)$ und + \[ + \kappa\colon x \mapsto \begin{cases} + x - \lfloor x\rfloor &x \in A\\ + x &\text{sonst} + \end{cases} + \] + Dann gilt $\forall c \in [0,1]\colon\kappa ^{-1}(c) = \{c, c+1, c-1,\dots\}$, + insbesondere ist $\kappa^{-1}(c)$ abzählbar und damit Element von $\mathscr B(\R)$. + Für $x \in A \cap [0,1]^c$ ist $\kappa(c)$ einfach die leere Menge. + Für $x\in A^c \cap [0,1]^c$ ist $\kappa(c) = \{c\}$. Damit ist die Bedingung an $\kappa$ erfüllt. + Dennoch ist $\kappa^{-1}([0,1]) = A$ und $A \notin \mathscr B(\R)$. + \end{enumerate} +\end{aufgabe} + \end{document}

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