# UE Stochastic Processes - Beispiel 1

Jump to: navigation, search

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} \,