1.1.2

Eine Markov-Kette ${X_{0}, X_{1}, ...}$ ${X_0,X_1, ...}$ mit Zustandsmenge $Z = {0, 1, 2}$ $Z = {0, 1, 2}$ habe die Übergangs-Matrix

$P = [\begin{matrix} 0.5 & 0 & 0.5 \\ 0.4 & 0.2 & 0.4 \\ 0 & 0.4 & 0.6 \end{matrix}]$ $P=[[0.5,0,0.5],[0.4,0.2,0.4],[0,0.4,0.6]]$

a) Bestimmen Sie die bedingten Wahrscheinlichkeiten $P (X_{2} = 2 ∣ X_{1} = 0, X_{0} = 1)$ $P(X_2 = 2 | X_1 = 0, X_0 = 1)$ sowie $P (X_{2} = 2, X_{1} = 0 ∣ X_{0} = 1)$ $P(X_2 = 2, X_ 1 = 0 | X_0 = 1)$ .

$P (X_{2} = 2 ∣ X_{1} = 0, X_{0} = 1) = P (X_{2} = 2 ∣ X_{1} = 0) = p_{0}, 2 = 0.5$ $P(X_2 = 2 | X_1 = 0, X_0 = 1)=P(X_2= 2|X_1 = 0)=p_0,2=0.5$

$P (X_{2} = 2, X_{1} = 0 ∣ X_{0} = 1) = p_{1}, 0 \cdot p_{1}, 2 = 0.4 \cdot 0.4 = 0.16$ $P(X_2 = 2, X_ 1 = 0 | X_0 = 1) = p_1,0*p_1,2 = 0.4 * 0.4= 0.16$

b) Die Anfangsverteilung sei $P (X_{0} = 0) = 0.4$ $P(X_0 = 0) = 0.4$ , $P (X_{0} = 1) = 0, 3$ $P(X_0 = 1) = 0,3$ , $P (X_{0} = 2) = 0, 3$ $P(X_0 = 2) = 0,3$ . Bestimmen Sie die Wahrscheinlichkeiten $P (X_{1} = 2)$ $P(X_1 = 2)$ und $P (X_{1} = 2, X 2 = 1)$ $P(X_1 = 2, X 2 = 1)$ !

$P (X_{1} = 2) = p_{0}, 2 \cdot P (X_{0} = 0) + p_{1}, 2 \cdot P (X_{0} = 1) + p_{2}, 2 \cdot P (X_{0} = 2)$ $P(X_1 = 2)=p_0,2 * P(X_0 = 0) + p_1,2 * P(X_0 = 1) + p_2,2 * P(X_0 = 2)$

$= 0.5 \cdot 0.4 + 0.3 \cdot 0.4 + 0.6 \cdot 0.3 = 0.5$ $= 0.5 * 0.4 + 0.3 * 0.4 + 0.6 * 0.3 = 0.5$

c) Ist die Markov-Kette irreduzibel? Ist sie aperiodisch?

Ja.