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Part 4.1 (5 points) A Markov chain has the following transition matrix: [ 0 0 1/3 1/3 1/3 ] [ 1/2 1/2 0 0 0 ] [ 0 0 0 1/3 2/3 ] [ 0 0 1 0 0 ] [ 0 0 1 0 0 ] 1) (1+1 = 2 points) Does this Markov chain have a single recurrent class? Please justify your answer. 2) (1 + 2 = 3 points) Does this Markov chain have a steady state and why? Part 4.2 (5 points) Consider a Markov chain with the following transition matrix: [ 2/5 1/5 1/5 1/5 ] [ 0 1/2 0 1/2 ] [ 0 1/3 2/3 0 ] [ 0 0 3/4 1/4 ] 1) (2 points) If the initial distribution of the 4 states S1, S2, S3 and S4 is v0 = <1, 0, 0, 0>, what is the probability that the chain is in state S2 after 2 steps. 2) (3 points) This Markov chain has a steady state. Compute the steady state probabilities. (Note that linear algebra methods are allowed to solve the equations).

          Part 4.1 (5 points) A Markov chain has the following transition matrix:

[ 0 0 1/3 1/3 1/3 ]
[ 1/2 1/2 0 0 0 ]
[ 0 0 0 1/3 2/3 ]
[ 0 0 1 0 0 ]
[ 0 0 1 0 0 ]

1) (1+1 = 2 points) Does this Markov chain have a single recurrent class? Please justify your answer.
2) (1 + 2 = 3 points) Does this Markov chain have a steady state and why?

Part 4.2 (5 points) Consider a Markov chain with the following transition matrix:

[ 2/5 1/5 1/5 1/5 ]
[ 0 1/2 0 1/2 ]
[ 0 1/3 2/3 0 ]
[ 0 0 3/4 1/4 ]

1) (2 points) If the initial distribution of the 4 states S1, S2, S3 and S4 is v0 = <1, 0, 0, 0>, what is the probability that the chain is in state S2 after 2 steps.
2) (3 points) This Markov chain has a steady state. Compute the steady state probabilities. (Note that linear algebra methods are allowed to solve the equations).

Part 4.1 (5 points) A Markov chain has the following transition matrix:

[ 0 0 1/3 1/3 1/3 ]
[ 1/2 1/2 0 0 0 ]
[ 0 0 0 1/3 2/3 ]
[ 0 0 1 0 0 ]
[ 0 0 1 0 0 ]

1) (1+1 = 2 points) Does this Markov chain have a single recurrent class? Please justify your answer.
2) (1 + 2 = 3 points) Does this Markov chain have a steady state and why?

Part 4.2 (5 points) Consider a Markov chain with the following transition matrix:

[ 2/5 1/5 1/5 1/5 ]
[ 0 1/2 0 1/2 ]
[ 0 1/3 2/3 0 ]
[ 0 0 3/4 1/4 ]

1) (2 points) If the initial distribution of the 4 states S1, S2, S3 and S4 is v0 = <1, 0, 0, 0>, what is the probability that the chain is in state S2 after 2 steps.
2) (3 points) This Markov chain has a steady state. Compute the steady state probabilities. (Note that linear algebra methods are allowed to solve the equations).

Added by Deborah T.

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