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61 lines
2.3 KiB
TeX
61 lines
2.3 KiB
TeX
\definition \textbf{Grundraum} $\Omega\qquad$ \textbf{Elementarereignis} $\omega \in \Omega$
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\definition \textbf{$\sigma$-Algebra} $\quad \mathcal{F} \subseteq \mathcal{P}(\Omega)\qquad$ \textbf{Ereignis} $A \in \mathcal{F}$
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\begin{tabular}{lll}
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(i) & $\Omega \in \mathcal{F}$ \\
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(ii) & $A \in \mathcal{F}$ & $\implies A^\comp \in \mathcal{F}$ \\
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(iii) & $A_1,\cdots,A_n \in \mathcal{F}$ & $\implies \displaystyle\underset{i \leq n}{\bigcup} A_i \in \mathcal{F}$
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\end{tabular}
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\lemma \textbf{Abgeschlossenheit} der $\sigma$-Algebra $\F$
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\begin{tabular}{ll}
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(i) & $\emptyset \in \F$ \\
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(ii) & $A_1,\cdots,A_n \in \F \implies \displaystyle\overunderset{\infty}{i=1}{\bigcap} A_i \in \F$ \\
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(iii) & $A, B \in \F \implies A \cup B \in \F$ \\
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(iv) & $A, B \in \F \implies A \cap B \in \F$
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\end{tabular}
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\definition \textbf{Wahrscheinlichkeitsmass} auf $(\Omega, \F): \P$
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$$
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\P: \F \to [0,1] \qquad \text{s.d.} \qquad A \mapsto \P[A]
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$$
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\begin{tabular}{ll}
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(i) & $\P[\Omega] = 1$ \\
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(ii) & $\P[A] = \displaystyle\sum_{i=1}^{\infty} A_i \iff A = \overunderset{\infty}{i=1}{\bigcup}A_i \quad \text{s.d.} \quad \overunderset{\infty}{i=1}{\bigcap}A_i = \emptyset$
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\end{tabular}
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\lemma \textbf{Eigenschaften} von $\P$
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\begin{tabular}{ll}
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(i) & $\P[\emptyset] = 0$ \\
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(ii) & $\displaystyle\overunderset{k}{i=1}{\bigcap} A_i = \emptyset \implies \P\biggl[ \overunderset{k}{i=1}{\bigcup} A_i \biggr] = \sum_{i=1}^{k} \P[A_i]$ \\
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(iii) & $\P[A^\comp] = 1 - \P[A]$ \\
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(iv) & $\P[A \cup B] = \P[A] + \P[B] - \P[A \cap B]$
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\end{tabular}
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\definition \textbf{Wahrscheinlichkeitsraum} $(\Omega, \F, \P)$
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\subtext{$A \in \mathcal{F}, \quad \omega \in \Omega$}
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\begin{tabular}{lll}
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$A$ tritt ein &$\iffdef$& $\omega \in A$ \\
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$A$ tritt nicht ein &$\iffdef$& $\omega \notin A$
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\end{tabular}
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\footnotesize
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\begin{tabular}{ll}
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(i) & $\emptyset$ tritt nie ein \\
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(ii) & $\Omega$ tritt immer ein
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\end{tabular}
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\normalsize
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\newpage
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\definition \textbf{Laplace Modell} $(\Omega, \F, \P)$
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\subtext{$\Omega$ endlich.}
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\begin{tabular}{ll}
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(i) & $\F = \mathcal{P}(\Omega)$ \\
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(ii) & $\forall A \in \F:\quad \P[A] = \displaystyle\frac{|A|}{|\Omega|}$
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\end{tabular}
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