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$\mathbb{P}\left( \sum_{n \in \N} |X_n| = \infty \right) = 0$ und + damit + \[ + \mathbb{P}\left( \sum_{n \in \N} |X_n| < \infty \right) = 1 + .\] + \item Es ist analog zu (i) + \[ + \E\left( \left| \sum_{n \in \N} X_n \right| \right) + \le \E\left( \sum_{n \in \N} |X_n| \right) + = \sum_{n \in \N} \E(|X_n|) < \infty + .\] Also $\sum_{n \in \N} X_n \mathscr{L}_1$ und $\sum_{n \in \N} |X_n| \in \mathscr{L}_1$. + \item Setze $S_n \coloneqq \sum_{k=1}^{n} X_k$. Es gilt + $\lim_{n \to \infty} S_n = \sum_{n \in \N} X_k \in \overline{\mathcal{A}}$ und + \[ + |S_n| = \left| \sum_{k=1}^{n} X_k \right| \le \sum_{k=1}^{n} |X_k| + \le \sum_{n \in \N} |X_n| \in \mathscr{L}_1 + .\] Insbesondere folgt $\sup_{n \in \N} |X_n| \le \sum_{n \in \N} |X_n| \in \mathscr{L}_1$. + + Wegen Monotonie der Erwartung + \[ + \E(|S_n|) \le \E\left( \sum_{k \in \N} |X_k| \right) + = \sum_{k \in \N} \E(|X_k|) < \infty + .\] + Also ist $S_n \in \mathscr{L}_1$ für $n \in \N$. Damit folgt mit dominierter Konvergenz im letzten + Schritt: + \begin{salign*} + \sum_{n \in \N} \E(X_n) + &= \lim_{n \to \infty} \sum_{k=1}^{n} \E(X_n) \\ + &\stackrel{\text{Linearität}}{=} \lim_{n \to \infty} \E\left( \sum_{k=1}^{n} X_n \right) \\ + &= \lim_{n \to \infty} \E(S_n) \\ + &= \E\left( \sum_{n \in \N} X_n \right) + .\end{salign*} + \end{enumerate} +\end{aufgabe} + +\stepcounter{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Es ist + \begin{salign*} + & \quad\qquad\var(X) = \E\left[ (X - \E(X))^2 \right] = 0 \\ + \stackrel{(X- \E(X))^2 \in \overline{\mathcal{A}}^{+}}{\iff}& + 1 = \mathbb{P}\left( (X - \E(X))^2 = 0 \right) = \mathbb{P}\left( X - \E(X) = 0 \right) + = \mathbb{P}( X = \E(X)) + .\end{salign*} + \item + \begin{itemize} + \item Es gilt nach Definition + \begin{salign*} + \cov(X,Y) &= \E \left[ (X - \E(X))(Y - \E(Y)) \right] \\ + &= \E \left[ (Y - \E(Y)) (X - \E(X)) \right] \\ + &= \cov(Y,X) + .\end{salign*} + \item Mit Linearität der Erwartung folgt direkt + \begin{salign*} + \cov(aX + bY, Z) &= \E\left[ (aX + bY - \E(aX + bY)(Z - \E(Z)) \right] \\ + &= \E\left[ (a(X - \E(X)) + b(Y - \E(Y)))(Z - \E(Z)) \right] \\ + &= a\E[ (X - \E(X))(Z - \E(Z)) ] + b \E[(Y - \E(Y))(Z - \E(Z))] \\ + &= a \cov(X, Z) + b \cov(Y, Z) + .\end{salign*} + \item Mit Monotonie der Erwartung im letzten Schritt folgt + \begin{salign*} + \cov(X, X) = \E[(X - \E(X))(X - \E(X))] = \E[\underbrace{(X - \E(X))^2}_{\ge 0}] \ge 0 + .\end{salign*} + \item Es gilt $\E(a) = a$, also + \begin{salign*} + \cov(a, X) = \E\left[ (a - \E(a))(X - \E(X)) \right] = \E(0) = 0 + .\end{salign*} + \end{itemize} + \item + \begin{itemize} + \item Mit der Linearität der Kovarianz und der letzten Eigenschaft in (b) folgt sofort + \begin{salign*} + \var(aX + b) &= \cov(aX + b, aX + b) \\ + &\stackrel{\text{linear}}{=} a \cov(X, aX + b) + \underbrace{\cov(b, aX + b)}_{= 0 \text{ (b.4)}} \\ + &\stackrel{\text{linear}}{=} a^2 \cov(X, X) + \underbrace{\cov(X, b)}_{= 0\text{ (b.4)}} \\ + &= a^2\var(X) + .\end{salign*} + \item Mit Linearität und Symmetrie folgt + \begin{salign*} + \var(X + Y) &= \cov(X + Y, X + Y) \\ + &= \cov(X, X) + \cov(X, Y) + \cov(Y, X) + \cov(Y, Y) \\ + &= \var(X) + \var(Y) + 2 \cov(X, Y) + .\end{salign*} + \end{itemize} + \end{enumerate} +\end{aufgabe} + +\end{document}