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1. (a) Consider a discrete-time Markov chain X = {X₂: ne N} with a transition matrix 0 1/2 1/2 M = 2/3 0 1/3 1/3 2/3 0 i. Draw the transition diagram of X. ii. Specify the period of each state - you are welcome to use R. Is the Markov chain irreducible, positive recurrent and ergodic? Justify your arguments. iii. Determine the stationary distribution of X from the equation лP = 1.

1. (a) Consider a discrete-time Markov chain X = {X₂: ne N} with a transition matrix 0 1/2 1/2 M = 2/3 0 1/3 1/3 2/3 0 i. Draw the transition diagram of X. ii. Specify the period of each state - you are welcome to use R. Is the Markov chain irreducible, positive recurrent and ergodic? Justify your arguments. iii. Determine the stationary distribution of X from the equation лP = 1.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
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1. (a) Consider a discrete-time Markov chain X = {Xn : n E N} with a transition matrix M = 0 1/2 1/2 2/3 0 1/3 1/3 2/3 0 i. Draw the transition diagram of X. ii. Specify the period of each state - you are welcome to use R. Is the Markov chain irreducible, positive recurrent and ergodic? Justify your arguments. iii. Determine the stationary distribution of X from the equation лP = π. π (b) Let X = {X, : t≥ 0} be a time-homogeneous Markov process with state-space S = {1, 2, 3}. The average time X spends in each state 1, 2 and 3 is given respectively by 1/2, 1/3 and 1/6 unit of time. The probability P(XT j|Xo = i) of making an immediate jump from state i to state j, with (i, j) € S, is given by the matrix M (1) in the question 1(a) above. Note that T refers to the jump time of the Markov process. i. Show that the corresponding intensity matrix Q of the Markov process is Q = = -2 1 1 2 2 4 -6 !). -3 1 ii. Find the average time it takes the process to move from state 1 to state 3.
Transcribed Image Text:1. (a) Consider a discrete-time Markov chain X = {Xn : n E N} with a transition matrix M = 0 1/2 1/2 2/3 0 1/3 1/3 2/3 0 i. Draw the transition diagram of X. ii. Specify the period of each state - you are welcome to use R. Is the Markov chain irreducible, positive recurrent and ergodic? Justify your arguments. iii. Determine the stationary distribution of X from the equation лP = π. π (b) Let X = {X, : t≥ 0} be a time-homogeneous Markov process with state-space S = {1, 2, 3}. The average time X spends in each state 1, 2 and 3 is given respectively by 1/2, 1/3 and 1/6 unit of time. The probability P(XT j|Xo = i) of making an immediate jump from state i to state j, with (i, j) € S, is given by the matrix M (1) in the question 1(a) above. Note that T refers to the jump time of the Markov process. i. Show that the corresponding intensity matrix Q of the Markov process is Q = = -2 1 1 2 2 4 -6 !). -3 1 ii. Find the average time it takes the process to move from state 1 to state 3.
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