UE Stochastic Processes - Beispiel 1

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Man zeige

P(A \cap B|C) = P(A|C) \cdot P(B|A \cap C) \,

beziehungsweise

P(X_2=i_2; X_1=i_1|X_0=i_0) = P(X_1=i_1|X_0=i_0) \cdot P(X_2 = i_2|X_0=i_0, X_1=i_1)\,

Def. von bedingter Wahrscheinlichkeit: P(A|B) = \frac{P(A \cap B)}{P(B)} \,

Daher 
 \begin{align} P(A|C) \cdot P(B|A \cap C) 
& = \frac{P(A \cap C)}{P(C)} \frac{P(A \cap B \cap C)}{P(A \cap C)} \\
& = \frac{P(A \cap B \cap C)}{P(C)} \\
& = P(A \cap B|C)
\end{align}
\,