Problem 4. Consider a Markov process that transit between states 0,1 and 2, with thetransition probability matrixP =0 1 20 1 0 01 0.1 0.6 0.32 0 0 1From this matrix we see that once the process reaches state 0 or state 2, it stays thereforever. Hence these two states are called absorbing states. If the process starts at state 1,the process may stay in state 1 for a duration but eventually the process will leave state1 and enter one of the absorbing states.1. What is the probability that the process twill enter into state 2 given the processstarts at state 1?2. On average how long does it take to reach one of the absorbing states?
Question
Problem 4. Consider a Markov process that transit between states 0,1 and 2, with thetransition probability matrixP =0 1 20 1 0 01 0.1 0.6 0.32 0 0 1From this matrix we see that once the process reaches state 0 or state 2, it stays thereforever. Hence these two states are called absorbing states. If the process starts at state 1,the process may stay in state 1 for a duration but eventually the process will leave state1 and enter one of the absorbing states.1. What is the probability that the process twill enter into state 2 given the processstarts at state 1?2. On average how long does it take to reach one of the absorbing states?
Solution
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To find the probability that the process will enter into state 2 given the process starts at state 1, we need to look at the transition probability matrix. From state 1, the probability to transition to state 2 is 0.3. Therefore, the probability that the process will enter into state 2 given the process starts at state 1 is 0.3.
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To find the average time it takes to reach one of the absorbing states, we need to calculate the expected number of steps to absorption. This can be done by solving the system of linear equations defined by (I-Q)T = 1, where Q is the submatrix of the transition matrix P excluding the rows and columns for the absorbing states. In this case, Q is a 1x1 matrix with the value 0.6. Solving the equation gives T = 1/(1-0.6) = 2.5. Therefore, on average, it takes 2.5 steps to reach one of the absorbing states.
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