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Wahrscheinlichkeitsraum $\Omega = \{(i_1, \dots, i_n)|i_j \in \{1,\dots, m\}\}$ als die Menge aller $n$-Tupel mit Werten zwischen 1 und $m$, wobei das $j$-te Element eines Tupels angibt, welche Ente der $j$-te Jäger gewählt hat. Dann enthält das Ereignis + \begin{align*} + A_i \coloneqq \{(i_1, \dots, i_n)\in \Omega, i\neq i_l \forall l \in \{1,\dots, n\}\} + \end{align*} + alle Elementarereignisse, in denen die $i$-te Ente nicht getroffen wird. + Die Zufallsvariable $X_i \colon \Omega \to \{0,1\}, \omega \mapsto \mathbbm{1}_{A_i}$ gibt an, ob die $i$-te Ente überlebt (1) oder nicht (0). Dann ist durch $X \coloneqq \sum_{i = 1}^{m} X_i$ gerade die Anzahl der überlebenden Enten gegeben. + Es gilt aufgrund der Linearität des Erwartungswerts + \begin{align*} + \E(X) &= \E\left(\sum_{i = 1}^{m} X_i\right)\\ + &= \sum_{i = 1}^{m} \E(X_i)\\ + &= \sum_{i = 1}^{m} \E(\mathbb{1}_{A_i})\\ + &= \sum_{i = 1}^{m} \P(A_i)\\ + &= \sum_{i = 1}^{m} \frac{\# A_i}{\# \Omega}\\ + &= \sum_{i = 1}^{m} \frac{(m-1)^n}{m^n}\\ + &= m \cdot \left(\frac{m-1}{m}\right)^n + \end{align*} + \item Wir bestimmen zunächst + \begin{align*} + \E(X_iX_j) &= \E(\mathbbm{1}_{A_i} \cdot \mathbbm{1}_{A_j})\\ + &= \E(\mathbbm{1}_{A_i \cap A_j})\\ + &= \P(A_i \cap A_j)\\ + \intertext{Für $i = j$ gilt $\P(A_i \cap A_j) = \P(A_i) = m\left(\frac{m-1}{m}\right)^n$. Sei also $i\neq j$} + &= \frac{\# A_i \cap A_j}{\# \Omega}\\ + &= \left(\frac{m-2}{m}\right)^2 + \end{align*} + Es gilt daher + \begin{align*} + \var(X) &= \E(X^2) - \E(X)^2\\ + &= \E\left(\sum_{i = 1}^{m} X_i \sum_{j = 1}^{m} X_j\right)- \E(X)^2\\ + &= \E\left(\sum_{i, j = 1}^m X_iX_j\right)- \E(X)^2\\ + &= \sum_{i,j = 1}^{m} \E(X_iX_j)- \E(X)^2\\ + &= \sum_{i = 1}^{m} \E(X_iX_i) + \sum_{i \neq j, 1\leq i, j \leq m} \E(X_iX_j)- \E(X)^2\\ + &= m \cdot \left(\frac{m-1}{m}\right)^n + (m^2 - m) \left(\frac{m-2}{m}\right)^2 - m^2 \cdot \left(\frac{m-1}{m}\right)^{2n} + \end{align*} + \item Für $n = m = 50$ gilt $7^{-2}\var(X) \approx 0.0996$ und $\E(X) \approx 18.2$. Für $m_1 = 11, m_2 = 26$ erhalten wir + \begin{align*} + \P(X \in [m_1, m_2]) &\geq \P(|X - \E(X)| \leq 7)\\ + &= 1 - \P(|X - \E(X)| > 7)\\ + \intertext{Ungleichung von Tschebycheff} + &\geq 1 - 7^{-2}\var(X)\\ + &\geq 1 - 0.0996\\ + &\geq 0.9 + \end{align*} + \end{enumerate} +\end{aufgabe} + \end{document}