let X be a markov chain with state space S=(1,2,3) please solve asap QUESTIONS Q

Question

let X be a markov chain with state space S=(1,2,3)

please solve asap

QUESTIONS Question 1 Let X be a Markov chain with state space S {1,2,3), initial proba- bility vector (,0,0), and one step transition matrix 0.6 0.1 04 P 06 0.2 0.2 0.4 0.1 0.5 Compute the following: (a) Pr(X 2); b) Pr(X (c) Pr(X2 2); d) Pr(X: 2, X 2IXe Pr(X: 2X. 2, Xi 2,Xu Question 2 For the following transition matrices determine the communicat- ing classes (and whether they are open or closed), absorbing states, transient and positive recurrent states (a) 1/2 0 0 1/2 0 1/2 1/2 0 0 0 P- 1/4 1/2 0 0 1/4 0 0 0 1 0 0 1/2 1/4 1/4 (b) 0 1/3 0 1/3 1/3 1/3 1/3 0 1/3 0 P 0 0 2/3 0 1/3 1/4 1/4 0 1/4 1/4 0 0 1/3 0 2/3

Explanation / Answer

(a) Pr(X1=2) = Pr(X1=2 | X0=1) = 0.1

(b) Pr(X2=2 | X0=1) = Pr(X1=1) * Pr(X2=2 | X1=1, X0=1) + Pr(X1=2) * Pr(X2=2 | X1=2, X0=1) + Pr(X1=3) * Pr(X2=2 | X1=3, X0=1)
= 0.5*0.1 + 0.1*0.2 + 0.4*0.1 = 0.11

(c) Pr(X2=2) = Pr(X2=2 | X0=1) = 0.11

(d) Pr(X2=2, X1=2 | X0=1) = Pr(X2=2, X1=2, X0=1) = 0.1 * 0.2 = 0.02

(e) Pr(X3=2 | X2=2, X1=2, X0=1) = Pr(X3=2 | X2=2) = 0.2

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