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$i$-ten Runde Zahl geworfen wird. Offenbar ist $k_i$ Bernoulli-verteilt mit + $n = 1$, $p = 0.5$. Daher gilt $R_i \overset{i.i.d}{\sim} 1 + (-0.5)^{B_{0.5}}$. + \item Es gilt + \begin{salign*} + \E(K_n) &= \E\left(\prod_{i = 1}^n R_i\right)\\ + &\stackrel{\indep R_i}{=} \prod_{i=1}^n \E(R_i)\\ + &\stackrel{R_i \sim 1 + (-0.5)^{B_{0.5}}}{=} \E(1 + (-0.5)^{B_{0.5}})^n\\ + &= (0.5(1.5 + 0.5))^n\\ + &= 1 + \end{salign*} + \item Es gilt + \begin{align*} + \E(\ln R_1) = 0.5 \cdot \ln(0.5) + 0.5 \cdot \ln(1.5) = \frac{\ln(3)}{2} - \ln(2) < 0. + \end{align*} + Insbesondere ist $\ln(R_1) \in \mathscr L_1$. Außerdem ist $(\ln R_i)_{i\in \N}$ eine unabhängig und identisch verteilte + Folge reeller Zufallsvariablen. Daher sind die Voraussetzungen fürs SGGZ erfüllt und wir erhalten + \begin{salign*} + \lim\limits_{n \to \infty} \frac{1}{n} \ln K_n &= \lim\limits_{n \to \infty} \frac{1}{n} \sum_{i = 1}^{\infty} \ln R_i\\ + &= \lim\limits_{n \to \infty} \overline{\ln R_i}\\ + &\stackrel{\text{SGGZ}}{=} \E(R_1)\\ + &= \frac{\ln(3)}{2} - \ln(2) < 0. + \intertext{Wir schließen daraus} + \lim\limits_{n \to \infty} \ln K_n &= -\infty. + \intertext{Schlussendlich folgt} + \lim\limits_{n \to \infty} K_n &= \lim\limits_{n \to \infty} e^{\ln K_n}\\ + &\stackrel{\operatorname{exp} \text{ stetig}}{=} e^{\lim\limits_{n \to \infty} \ln K_n}\\ + &= 0. + \end{salign*} + \end{enumerate} \end{aufgabe} \begin{aufgabe}[] - + \begin{enumerate}[(a)] + \item Aus dem ZGWS folgt mit Korollar 31.04: $\sqrt{n}[\overline{X_n}-\mu] \xrightarrow{D} N_{(0,\sigma^2)}$. + Daraus erhalten wir für $a_n = \sqrt{n}, X_n = \overline{X_n}$ und $x = \mu$ mit Satz 29.19 + $\overline{X}_n \xrightarrow{\mathbbm{P}} \mu = \E(X_1)$. Das ist genau die Aussage des schwachen Gesetzes + der großen Zahlen. + \item Wir zeigen $X_n \xrightarrow{\mathbbm{P}} 0$. Daraus folgt dann die Behauptung. + + Sei $\epsilon > 0$. + Es gilt wegen $\sqrt[n]{n} \xrightarrow{n \to \infty} 1$ + \begin{align*} + \lim\limits_{n \to \infty} \sqrt[n]{\frac{2\epsilon}{n+1}} = 1. + \end{align*} + Für beliebiges $x \in (-1, 1)$ existiert daher ein $n\in \N$ mit $|x| < \sqrt[n]{\frac{2\epsilon}{n+1}}$. + Insbesondere ist + \begin{align*} + \limsup\limits_{n \to \infty} \left\{|\frac{n+1}{2} |x|^n \mathbbm{1}_{(-1,1)}| > \epsilon\right\} + &= \limsup\limits_{n \to \infty} \left\{x \in (-1, 1), |x| > \sqrt[n]{\frac{2\epsilon}{n+1}}\right\}= \emptyset + \end{align*} + und damit auch $\limsup\limits_{n \to \infty} \mathbbm{P}(\{|X_n| > \epsilon\}) = 0$. + Das ist gerade die Definition von $X_n \xrightarrow{\mathbbm{P}} 0$, die Behauptung ist also bewiesen. + \item \begin{enumerate} + \item Es gilt $\forall x < 0\colon \F_n(x) = 0 \implies \F(x) = 0$, + $\forall x \in [0,1]\colon \F_n(x) = \frac{x}{1 + 1/n} \implies \F(x) = x$ und + $\forall x \geq 1\colon \F_n(x) = \frac{x}{1+1/n}$ für $x < 1+ 1/n \Leftrightarrow n < \frac{1}{x-1}$. + Für $n > \frac{1}{x-1}$ gilt $\F_n(x) = 0$. Daraus folgt $\F(x) = 0$. + Wir erhalten als Grenzwert der Verteilungsfunktionen + \begin{align*} + \lim\limits_{n \to \infty} \F_n(x) = \F(x) \coloneqq \begin{cases} + 0 &| x < 0\\ + x &| 0 \leq x \leq 1\\ + 1 &| 1 < x + \end{cases} + \end{align*} + Dies ist genau die Verteilungsfunktion von $U_{[0,1]}$, also wähle gilt $X_n \xrightarrow{D} U_{[0,1]}$. + \item Es gilt + \begin{align*} + \lim\limits_{n \to \infty} \F_n(x) &= \lim\limits_{n \to \infty} \begin{cases} + \int_0^x ne^{-n\tilde{x}}\d{\tilde{x}} = 1 - e^{-nx} &| x > 0\\ + 0 &| x \leq 0 + \end{cases}\\ + &= \begin{cases} + 1 - \lim\limits_{n \to \infty} e^{-nx} &| x > 0\\ + 0&|x \leq 0 + \end{cases}\\ + &= \mathbbm{1}_{\R_{\setminus 0}^+} + \end{align*} + Es gilt aber $\lim\limits_{x \searrow 0} \mathbbm{1}_{\R_{\setminus 0}^+}(x) = 1 \neq 0 = \mathbbm{1}_{\R_{\setminus 0}^+}(0)$. + Daher ist der Grenzwert der Verteilungsfunktion nicht rechtsstetig. Es kann daher keine Zufallsvariable mit dieser Verteilung + existieren. + \item Es gilt + \begin{align*} + \lim\limits_{n \to \infty} \F_n(x) &= \lim\limits_{n \to \infty} \begin{cases} + \int_0^x \frac{1}{n}e^{-\frac{1}{n}\tilde{x}}\d{\tilde{x}} = 1 - e^{-\frac{1}{n}x} &| x > 0\\ + 0 &| x \leq 0 + \end{cases}\\ + &= \begin{cases} + 1 - \lim\limits_{n \to \infty} e^{-\frac{1}{n}x} = 0 &| x > 0\\ + 0&|x \leq 0 + \end{cases}\\ + &\equiv 0 + \end{align*} + Es gilt aber $\lim\limits_{x \to \infty} 0 = 0\neq 1$. Es kann daher keine Zufallsvariable mit dieser Verteilung + existieren. + \end{enumerate} + \end{enumerate} \end{aufgabe} \begin{aufgabe} @@ -45,7 +141,7 @@ es folgt für $t \in \R$ \[ \varphi_{Y-Z}(t) = \varphi_{Y} \varphi_{-Y}(t) = \varphi_Y(t) \overline{\varphi_Y(t)} - = |\varphi_Y(t)| \ge 0 + = |\varphi_Y(t)|^2 \ge 0 .\] Aber $\varphi_{X}(t) = \frac{\sin(t)}{t}$ für $X \sim U_{[-1, 1]}$ und für $t = \frac{3}{2} \pi$ folgt $\varphi_X(\frac{3}{2}\pi) = - \frac{2}{3 \pi} < 0$. Also ist $\varphi_X(t) \neq \varphi_{Y-Z}(t)$, also