answer all parts for exercise 9.2.4 and especially explain part c. and show all

Question

answer all parts for exercise 9.2.4 and especially explain part c. and show all work.

se 92.3 The tranobability matrix of a discre obability matrix of a discrete-time Markov chain is given by 'e random ote by Y, and buted, and are s an alternating r interest with at the process associating a 0 0 0 1.0 0 0 1.0 0 0 P=10001.0 0 0 0.8 0.2 00 0.4 0 0.6 0 0 w all sample paths of length 4 that begin in state 1. What is the probability of being in each of the states1 hrough 5 after four steps beginning in state 1? Evercise ransition probability matrix is given by 9.2.4 Consider a discrete-time Markov chain consisting of four states a, b, c, and d and whose own. As soon tioning state s the random orresponding ribed above, 0.0 0.0 1.0 0.0 0.0 0.4 0.6 0.0 0.8 0.0 0.2 0.0 0.2 0.3 0.0 0.5 ime that the emachine is Compute the following probabilities 9.2.5 at time step n, n A Markov chain with two states a and b has the following conditional probabilities: If it is in . 1,2, ..., then it stays in state a with probability 0.5(0.5)". If it is in state b at 0

Explanation / Answer

Question 9.2.4:

a) The path here is given as:

a --> c --> c --> c --> c, therefore the required probability here is computed as: Pac*Pcc*Pcc*Pcc = 1*0.2*0.2*0.2 = 0.008

Therefore 0.008 is the required probability here.

b) Here from step 3 to step 4 we are going from state a to state b, but there is no probability of going from state a to state b, therefore the required probability here is 0

c) Here we are already given that we are state d in time point 3 and then in state b in time point 4. Therefore the path here is given as:

b --> c --> a --> c --> c

Prob. = 0.6*0.8*1*0.2 = 0.096

Therefore 0.096 is the required probability here.

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