A Markov chain has the transition matrix shown below:[0.6 0.4P =[0.8 0.2](Note: For questions 1, 2, and 4, express your answers as decimal fractions rounded to 4 decimal places (if they have more than 4 decimal places).)(1) If, on the first observation the system is in state 1, what is the probability that it is in state 1 on the second observation?(2) If, on the first observation the system is in state 1, what is the probability that it is in state 1 on the third observation?(3) If, on the first observation, the system is in state 2, what state is the system most likely to occupy on the third observation? (If there is more than one such state, which is the first one.)(4) If, on the first observation, the system is in state 2, what is the probability that it alternates between states 1 and 2 for the first four observations (i.e., it occupies state 2, then state 1, then state 2, andfinally state 1 again)?

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Asked Dec 5, 2019
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A Markov chain has the transition matrix shown below:
[0.6 0.4
P =
[0.8 0.2]
(Note: For questions 1, 2, and 4, express your answers as decimal fractions rounded to 4 decimal places (if they have more than 4 decimal places).)
(1) If, on the first observation the system is in state 1, what is the probability that it is in state 1 on the second observation?
(2) If, on the first observation the system is in state 1, what is the probability that it is in state 1 on the third observation?
(3) If, on the first observation, the system is in state 2, what state is the system most likely to occupy on the third observation? (If there is more than one such state, which is the first one.)
(4) If, on the first observation, the system is in state 2, what is the probability that it alternates between states 1 and 2 for the first four observations (i.e., it occupies state 2, then state 1, then state 2, and
finally state 1 again)?
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A Markov chain has the transition matrix shown below: [0.6 0.4 P = [0.8 0.2] (Note: For questions 1, 2, and 4, express your answers as decimal fractions rounded to 4 decimal places (if they have more than 4 decimal places).) (1) If, on the first observation the system is in state 1, what is the probability that it is in state 1 on the second observation? (2) If, on the first observation the system is in state 1, what is the probability that it is in state 1 on the third observation? (3) If, on the first observation, the system is in state 2, what state is the system most likely to occupy on the third observation? (If there is more than one such state, which is the first one.) (4) If, on the first observation, the system is in state 2, what is the probability that it alternates between states 1 and 2 for the first four observations (i.e., it occupies state 2, then state 1, then state 2, and finally state 1 again)?

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Expert Answer

Step 1

1.

The given transition probability matrix can be represented as,

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ai a2 a¡|0.6 | 0.4 a2 0.8 0.2

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Step 2

The probability of transition for one state to another is,

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P(a, → a,) = 0.6 P(a, → a,) =0.4 P(a, → a,) = 0.8 P(a, →a, )= 0.2 If, on the first observation the system is in state 1, the probability that it is in state 1 on the second observation is 0.6. That is, P(a, → a,)= 0.6 %3D

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Step 3

2.

If, on the first observation the system is in state 1, the probab...

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[0.6 0.4T0.6 0.4] p? 0.8 0.2 || 0.8 0.2 0.68 0.32 0.64 0.36

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