From eebc8aaa76462b5e4a33775243836006b84a77ef Mon Sep 17 00:00:00 2001 From: flavis Date: Tue, 1 Dec 2020 19:12:12 +0100 Subject: [PATCH] add wtheo 4: ex 13 & 15, update lecture cls --- lecture.cls | 2 + wtheo4.pdf | Bin 0 -> 130145 bytes wtheo4.tex | 153 ++++++++++++++++++++++++++++++++++++++++++++++++++++ 3 files changed, 155 insertions(+) create mode 100644 wtheo4.pdf create mode 100644 wtheo4.tex diff --git a/lecture.cls b/lecture.cls index 904ada4..0a9796a 100644 --- a/lecture.cls +++ b/lecture.cls @@ -25,6 +25,8 @@ \RequirePackage{calc} \RequirePackage{wasysym} \RequirePackage{environ} +\RequirePackage{bbm} +\RequirePackage{mathrsfs} \usetikzlibrary{quotes, angles} \pgfplotsset{ diff --git a/wtheo4.pdf b/wtheo4.pdf new file mode 100644 index 0000000000000000000000000000000000000000..7bac902ae1dc5c774909726b3034c0784d1e497e GIT binary patch literal 130145 zcmb??bx_>f((Sc{6suWfHv%7ci?zL90AHABC3@Zm44+{PGbl*G*FBOQ&$df z&6dTCvyuFQBf3EHx~;N2NbWbn+DHaNe8e4 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\begin{proof} + \begin{enumerate}[(i)] + \item Es ist $\mathbb{P}^{X} \ge 0$, da $\mathbb{P} \ge 0$. + \item $\mathbb{P}^{X}(\mathcal{X}) = \mathbb{P}(X^{-1}(\mathcal{X})) = \mathbb{P}(\Omega) = 1$, da $\mathbb{P}$ W-Maß. + \item Zunächst ist für $A, B \subseteq \mathcal{X}$ mit $A \cap B = \emptyset$ auch $X^{-1}(A) \cap X^{-1}(B) =X^{-1}(A \cap B) = X^{-1}(\emptyset) = \emptyset$. + Also bleiben disjunkte Vereinigungen unter Urbildbildung disjunkt $(*)$. + + Seien nun $B_i \in \mathscr{B}$ für $i \in \N$ und paarweise verschieden. Dann folgt + \begin{salign*} + \mathbb{P}^{X}\left( \bigcupdot_{i \in \N} B_i \right) + &= \mathbb{P}\left( X^{-1}\left( \bigcupdot_{i \in \N} B_i \right) \right) \\ + &\stackrel{(*)}{=} \mathbb{P}\Big( \bigcupdot_{i \in \N} \underbrace{X^{-1}(B_i)}_{\in \mathscr{A}} \Big) \\ + &\stackrel{\mathbb{P} \text{ Maß}}{=} + \sum_{i \in \N} \mathbb{P}(X^{-1}(B_i)) \\ + &= \sum_{i \in \N} \mathbb{P}^{X}(B_i) + .\end{salign*} + \end{enumerate} + \end{proof} + \item Beh.: $\left( \mathbb{P}^{X} \right)^{Y} = \mathbb{P}(Y \circ X)$. + \begin{proof} + Sei $C \in \mathscr{C}$. + \begin{salign*} + Y^{-1}(X^{-1}(C)) &= \{ x \in \Omega \mid X(x) \in \{ y \in \mathcal{X} \mid Y(y) \in C\} \} \\ + &= \{ x \in \Omega \mid Y(X(x)) \in C\} \\ + &= (Y \circ X)^{-1}(C) + .\end{salign*} + Damit folgt + \[ + (\mathbb{P}^{X})^{Y}(C) = \mathbb{P}^{X}(Y^{-1}(C)) = \mathbb{P}(X^{-1}(Y^{-1}(C))) + = \mathbb{P}((Y \circ X)^{-1}(C)) = \mathbb{P}^{(Y \circ X)} + .\] + \end{proof} + \item Beh.: Es ist + \[ + \mathbb{P}^{X}(\{0\}) = \frac{4}{7} \qquad \mathbb{P}^{X}(\{1\}) = \frac{2}{7} + \qquad \mathbb{P}^{X}(\{2\}) = \frac{1}{7} + .\] Damit ist $\mathbb{P}^{X}$ eindeutig festgelegt. + \begin{proof} + Es ist $\text{Bild}(X) = \{0, 1, 2\}$. Damit ist + $(\text{Bild}(X), 2^{\text{Bild}(X)}, \mathbb{P}^{X})$ diskreter + Wahrscheinlichkeitsraum. Es genügt also $\mathbb{P}^{X}$ für + alle Elementarereignisse zu bestimmen. + + Damit folgt mit geometrischer Reihe + \begin{salign*} + \mathbb{P}^{X}(\{0\}) &= \mathbb{P}(X^{-1}(\{0\})) + = \mathbb{P}(3 \N_0) = \sum_{k \in \N_0} 2^{-3k-1} + = \frac{1}{2} \sum_{k \in \N_0} \left( \frac{1}{8} \right)^{k} + = \frac{1}{2} \frac{1}{1 - \frac{7}{8}} = \frac{4}{7} \\ + \mathbb{P}^{X}(\{1\}) &= \mathbb{P}(3 \N_0 + 1) + = \sum_{k \in \N_0} 2^{-3k-1-1} = \frac{2}{7} \\ + \mathbb{P}^{X}(\{2\}) &= \mathbb{P}(3 \N_0 + 2) + = \sum_{k \in \N_0} 2^{-3k-2-1} = \frac{1}{7} + .\end{salign*} + \end{proof} + \end{enumerate} +\end{aufgabe} + +\stepcounter{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Beh.: $\forall y \in [0,1], z \in \R$ gilt $\mathbb{F}^{*}(y) \le z \iff y \le \mathbb{F}(z)$. + \begin{proof} + Sei $y \in [0,1]$ und $z \in \R$. + \begin{itemize} + \item ,,$\implies$''. Sei also $\mathbb{F}^{*}(y) \le z$. + Da $\mathbb{F}$ monoton wachsend, folgt direkt + $\mathbb{F}(\mathbb{F}^{*}(y)) \le \mathbb{F}(z)$. + + Also genügt es z.z.: $y \le \mathbb{F}(\mathbb{F}^{*}(y))$. + Betrachte dazu $x_n \coloneqq \mathbb{F}^{*}(y) + \frac{1}{n}$ für $n \in \N$. Nach + der Definition von $\mathbb{F}^{*}(y)$ folgt $\mathbb{F}(x_n) \ge y$ $\forall n \in \N$. + Außerdem gilt $x_n \downarrow \mathbb{F}^{*}(y)$ für $n \to \infty$. Mit + der Rechtsstetigkeit von $\mathbb{F}$ folgt damit + $\mathbb{F}(x_n) \downarrow \mathbb{F}(\mathbb{F}^{*}(y))$. + + Das heißt für $\epsilon > 0$ ex. ein $n_0 \in \N$, s.d. $\forall n \ge n_0$ gilt, dass + $|\mathbb{F}(x_n) - \mathbb{F}(\mathbb{F}^{*}(y))| < \epsilon$. Da + $\mathbb{F}$ monoton wachsend und $x_n \ge \mathbb{F}^{*}(y)$ folgt + \begin{salign*} + \mathbb{F}(x_n) &= \mathbb{F}(\mathbb{F}^{*}(y)) + \epsilon + \intertext{Also da $y \le \mathbb{F}(x_n)$ $\forall n \in \N$} + y &\le \mathbb{F}(\mathbb{F}^{*}(y)) + \epsilon + .\end{salign*} + Mit $\epsilon \to \infty$ folgt $y \le \mathbb{F}(\mathbb{F}^{*}(y))$ und damit die + Behauptung. + \item ,, $\impliedby$'': Sei also $y \le \mathbb{F}(z)$. Dann + folgt direkt + \[ + \mathbb{F}^{*}(y) = \inf \{ x \in \R \mid \mathbb{F}(x) \ge y\} \le z + .\] + \end{itemize} + \end{proof} + \item Beh.: Ist $Y \sim U[0,1]$ dann hat $\mathbb{F}^{*}(Y)$ dieselbe Verteilung wie $X$. + \begin{proof} + Sei $Y \sim U[0,1]$. Dann ist $Y(\omega) \in [0,1]$ $\forall \omega \in \Omega$ und + es folgt für $z \in \R$ aus (a), dass + $\mathbb{F}^{*}(Y(\omega)) \le z \iff Y(\omega) \le \mathbb{F}(z)$ $\forall \omega \in \Omega$ + und damit + \[ + \mathbb{F}^{*}(Y) \le z \iff Y \le \mathbb{F}(z) \quad (*) + .\] + Außerdem gilt für $y \in [0,1]$ da $Y \sim U[0,1]$ + \[ + \mathbb{P}(Y \le y) = y \qquad (**) + .\] + Damit folgt für $x \in \R$: + \[ + \mathbb{P}(\mathbb{F}^{*}(Y) \le x) + \stackrel{(*)}{=} \mathbb{P}(Y \le \mathbb{F}(x)) + \; \stackrel{(**)}{=} \; \mathbb{F}(x) + .\] Also sind $\mathbb{F}^{*}(Y)$ und $\mathbb{F}$ identisch verteilt. + \end{proof} + \item Sei $\lambda > 0$. Beh.: + \[ + G(x) \coloneqq \begin{cases} + -\frac{1}{\lambda} \ln(1-x) & x \in [0,1) \\ + \infty & x = 1 + \end{cases} + .\] + \begin{proof} + Es ist $X \sim \text{Exp}_{\lambda}$. Also definiere + \begin{salign*} + \mathbb{F}\colon (0, \infty) &\to [0,1) \\ + x &\mapsto \mathbb{F}^{X}(x) = \mathbb{F}_{\text{Exp}_\lambda}(x) = 1 - \exp(-\lambda x) + .\end{salign*} + Dann ist $\mathbb{F}$ invertierbar und es gilt $\mathbb{F}^{*} = \mathbb{F}^{-1}$ + auf $(0,1)$. Weiter ist + \begin{salign*} + \mathbb{F}^{-1}(x) = -\frac{1}{\lambda} \ln(1-x) \qquad x \in [0,1) + .\end{salign*} + Wähle dann $G$ wie in Beh. Dann ist $G = \mathbb{F}^{*}$ auf $(0,1)$ und + $G(0) = 0 = \inf \{x \in \R^{+}_0 \mid \mathbb{F}(x) \ge 0\} = \mathbb{F}^{*}(0)$. + Außerdem gilt $\mathbb{F}^{*}(1) = \inf \{ x \in \R^{+}_0 \mid \mathbb{F}(x) = 1\} + = \infty = G(1)$. Damit folgt die Behauptung aus (b). + \end{proof} + \end{enumerate} +\end{aufgabe} + +\end{document}