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zLjCM*Q40#kQB=?cBvSkB&C!-nWYoR(G2D}OnAJ&T$nwFR_`Q^*1esgCwYQ&@G??QX zGHS~hg@F%harH4@P7oshiOdZL0_p%A{!CbK*G%)JPJg#IIDYQk5DVWa+}f!R_$L!h z`YHJd3Hb6-UbSw%ZFTax)$ifw60UXlTpbZ5`Av~5K^a<}rUL|I$veZaHf?9mm=WEh z@6%(Xbxw!*hvn{Nv6uMKU~)G2cNc#2pT>kP+pqx0QII2Dg^MV59Ux4n%`G@`e4mzw z%--IAhs$47`;C6ymUo%Z{=wtg+#;VKF4?;gGu%cz1GdN5ao_K+qv0H6Tcz=Pg50tJ zO@p)A?;2-*jcTn4!d@KDGIG zA_3ET$LTJwk!>+?xGphhZoF({cZYi@zHIfQf{09XBI9_9`09x`67YrzXW{q6I@>$Y zVGzyFWzirTh`T8vWNi&tz5yLwOC`%o%nPKAbMPvQN{sHu6d<3m8$<0y`@EX$XdYZ3 z|HT~ob4-rU^hR3x*Jd5vJ9~$c7bXL4WjD`GtBlr2aQo_$D!FF9%j2Rg6#MRhO}8U` zpEbs~c<=MGrlC#xzbiR$)DsLIm7`EQT&uMA&o6)Jk7GCg>T_$;-s{)1^-9_F@?F|( z>>OWJesF#a%soCwcS8R~G5M!BV&pwCS$=_XiB5BAW1~j7VK?hd1+Qy-eQU;NQMLFS zou=IMYvjJ4IZL8We>}WnYcMipy(Ta`l%aW&P=>HRcRAV&z(+Ikhv#bTi4#Oa2XKLm_q{B$W!LA3xpaSqR`?-T*&>+%bB4Qz8`l(a4w|61p_#u)Z zQvQ#)xjhlbe@rR=DTos3{5VY5McA2HL_|cG#krU{nAw=bg_t=xMZ|?USh$#kSow+m z&mupg$eP(%xLE#>*RcMVnc@G!&VWDF$sDYN7Pf!8IpZiPd?1VGL^PBakCieLJ#VTT zE>160#C!;{6=0N1U9DfKa^Va*)tPCr(`{Z&C*?0i9XALj8zhwaP}oriNl^zB9xnu& zITvQa83ei(9; n$. + + Dann ist $(A_n)_{n \in \N}$ eine Partition von $\Omega$. Damit folgt + mit Satz von der totalen Wahrscheinlichkeit: + \begin{salign*} + \mathbbm{p}(k) &= \mathbb{P}(A_n) \\ + &= \mathbb{P}(A_n \cap \Omega) \\ + &= \sum_{n \in \N_0} \mathbb{P}(A_n) \mathbb{P}(B_k \mid A_n) \\ + &= \sum_{n \in \N_0} \frac{\lambda^{n}}{n!} e^{-\lambda} \binom{n}{k} p^{k} (1-p)^{n-k} \\ + &= e^{-\lambda} p^{k} \sum_{n=k}^{\infty} \frac{\lambda^{k}\lambda^{n-k}}{n!} \frac{n!}{k!(n-k)!} (1-p)^{n-k} \\ + &= e^{-\lambda} \frac{(\lambda p)^{k}}{k!} \sum_{n=k}^{\infty} \frac{\lambda^{n-k}}{(n-k)!} + (1-p)^{n-k} \\ + &= e^{-\lambda} \frac{(\lambda p)^{k}}{k!} \sum_{n=0}^{\infty} \frac{\lambda^{n}}{n!} + (1-p)^{n} \\ + &= e^{-\lambda} \frac{(\lambda p)^{k}}{k!} e^{\lambda(1-p)} \\ + &= e^{-\lambda p} \frac{(\lambda p)^{k}}{k!} + .\end{salign*} + \end{proof} + \item Seien $X$ bzw. $Y$ die Lebensdauer in Tagen von Lampe 1 bzw. Lampe 2 mit + $X \sim \text{Poi}_{\lambda_1}$ und $Y \sim \text{Poi}_{\lambda_2}$. Nach Vorraussetzung + ist $X \indep Y$, also + $X + Y \sim \text{Poi}_{\lambda_1} * \text{Poi}_{\lambda_2}$. + Nach VL hat $\text{Poi}_{\lambda_1} * \text{Poi}_{\lambda_2}$ die Zähldichte + \begin{salign*} + (\mathbbm{p}_1 * \mathbbm{p}_2)(n) + &= \sum_{k=0}^{n} \mathbbm{p}_1(n-k) \mathbbm{p}_2(k) \\ + &= \sum_{k=0}^{n} \frac{\lambda_1^{n-k}}{(n-k)!} e^{- \lambda_1} + \frac{\lambda_2^{k}}{k!} e^{- \lambda_2} \\ + &= e^{-(\lambda_1 + \lambda_2)} \frac{1}{n!} \sum_{k=0}^{n} + \underbrace{\frac{n!}{(n-k)!k!}}_{= \binom{n}{k}} \lambda_1^{n-k} + \lambda_2^{k} \\ + &= e^{-(\lambda_1 + \lambda_2)} \frac{(\lambda_1 + \lambda_2)^{n}}{n!} + .\end{salign*} + Also folgt $X + Y \sim \text{Poi}_{\lambda_1 + \lambda_2}$. + \end{enumerate} +\end{aufgabe} \begin{aufgabe} \begin{enumerate}[(a)] @@ -47,7 +96,79 @@ \end{enumerate} \end{aufgabe} -\stepcounter{aufgabe} +\begin{aufgabe} + (i)-(iii) bezeichne im Folgenden die Eigenschaften von $\E$ aus Satz 20.01. + \begin{enumerate}[(a)] + \item Sei $(X_n)_{n \in \N}$ mit $X_n \in \overline{\mathscr{A}}^{+}$. Dann setze + $S_n \coloneqq \sum_{k=1}^{n} X_k$. Dann ist $S_n \in \overline{\mathscr{A}}^{+}$ und + $S_n \uparrow \sum_{n \in \N} X_n$. Damit folgt + \begin{salign*} + \E\left( \sum_{n \in \N} X_n\right) &\stackrel{\text{(ii)}}{=} + \lim_{n \to \infty} \E(S_n) \\ + &= \lim_{n \to \infty} \E\left( \sum_{k=1}^{n} X_n \right) \\ + &\stackrel{\text{(i)}}{=} + \lim_{n \to \infty} \sum_{k=1}^{n} \E(X_n) \\ + &= \sum_{n \in \N} \E(X_n) + .\end{salign*} + \item + \begin{itemize} + \item ,,$\implies$'': Sei $\indep_{i \in I} \mathcal{A}_i$ und + $(X_i)_{i \in I}$ mit $X_i \in \overline{\mathcal{A}}_i^{+}$ für $i \in I$. Sei weiter + $\emptyset \neq J \subseteq I$ endlich. + + \begin{enumerate}[(1)] + \item Seien $X_i$ Bernoulli ZV mit $X_i = \mathbbm{1}_{A_i}$ und + $A_i \in \mathcal{A}_i$ für $i \in I$. Da + $\indep_{i \in I} \mathcal{A}_i \implies \indep_{i \in I} A_i$. + Dann gilt + \begin{salign*} + \E\left( \prod_{j \in J}^{} X_j \right) + &= \E\left( \prod_{j \in J}^{} A_j \right) \\ + &= \E( \mathbbm{1}_{\bigcap_{j \in J} A_j} ) \\ + &\stackrel{\text{(iii)}}{=} \mathbb{P}(\bigcap_{j \in J} A_j) \\ + &\stackrel{\indep_{i \in I} A_i}{=} + \prod_{j \in J}^{} \mathbb{P}(A_j) \\ + &\stackrel{\text{(iii)}}{=} \prod_{j \in J}^{} \E(\mathbbm{1}_{A_j}) \\ + &= \prod_{j \in J}^{} \E(X_j) + .\end{salign*} + \item Seien nun $X_i$ einfache positive numerische ZV. Dann ist + $\prod_{j \in J}^{} X_j $ eine Summe von Produkten aus skalierten + Bernoulli-ZV. Mit (a), (ii) und Schritt (1) folgt die Behauptung für + $(X_i)_{i \in I}$. + \item Seien nun $X_i \in \overline{\mathcal{A}}_i^{+}$ für $i \in I$. Für + $i \in I$ ex. dann eine Folge $(X_{in})_{n \in \N}$ mit $X_{in} \uparrow X_i$ + und $X_{in}$ einfach positiv numerische ZV. + Da $J$ endlich folgt dann + \begin{salign*} + \prod_{i \in J} X_i = \prod_{i\in J}^{} \lim_{n \to \infty} + X_{in} = \lim_{n \to \infty} \prod_{i \in J}^{} X_{in} + .\end{salign*} + Also folgt $\prod_{j \in J} X_{in} \uparrow \prod_{j \in J}^{} X_i $. Damit folgt + \begin{salign*} + \E \left( \prod_{j \in J}^{} X_j \right) &= + \E\left( \prod_{j \in J}^{} \lim_{n \to \infty} X_{jn} \right) \\ + &= \E \left( \lim_{n \to \infty} \prod_{j \in J}^{} X_{jn} \right) \\ + &\stackrel{\text{(ii)}}{=} \lim_{n \to \infty} + \E\left( \prod_{j \in J}^{} X_{jn} \right) \\ + &\stackrel{\text{(2)}}{=} \lim_{n \to \infty} \prod_{j \in J}^{} \E(X_{jn}) \\ + &= \prod_{j \in J}^{} \lim_{n \to \infty} \E(X_{jn}) \\ + &\stackrel{\text{(ii)}}{=} \prod_{j \in J}^{} \E(X_j) + .\end{salign*} + \end{enumerate} + \item ,,$\impliedby$'': Sei $(A_i)_{i \in I}$ mit $A_i \in \mathcal{A}_i$ für $i \in I$. + Z.z.: $\indep_{ i \in I} A_i$. Dazu sei $\emptyset \neq J \subseteq I$ endlich. + Betrachte $X_i \coloneqq \mathbbm{1}_{A_i}$. Dann ist nach Vorraussetzung + \begin{salign*} + \mathbb{P}\left( \bigcap_{i \in J} A_i \right) + &\stackrel{\text{(iii)}}{=} + \E \left( \mathbbm{1}_{\bigcap_{i \in J} A_i} \right) \\ + &= \E\left( \prod_{i \in J}^{} \mathbbm{1}_{A_i} \right) \\ + &\stackrel{\text{Vorr.}}{=} \prod_{i \in J}^{} \E(\mathbbm{1}_{A_i}) \\ + &\stackrel{\text{(iii)}}{=} \prod_{i \in J}^{} \mathbb{P}(A_i) + .\end{salign*} + \end{itemize} + \end{enumerate} +\end{aufgabe} \begin{aufgabe} \begin{enumerate}[(a)]