Suppose a government study estimated that the probability of successive generations of a rural family remaining in a rural area was 0.4 and the probability of successive generations of an urban family remaining in an urban area was 0.6. Assuming a Markov chain applies to these facts, find the steady-state vector.

Question

Suppose a government study estimated that the probability of successive generations of a rural family remaining in a rural area was 0.4 and the probability of successive generations of an urban family remaining in an urban area was 0.6. Assuming a Markov chain applies to these facts, find the steady-state vector. ​

Quizplus · Accepted Answer

To find the steady-state vector, we set up a system of equations where the probabilities of each state in the long term are equal to the probabilities of transitioning to that state from the other states in the long term. Letting R be the probability of being in a rural area and U be the probability of being in an urban area, we have:

R = 0.4R + 0.6U
U = 0.6R + 0.4U

Solving for R and U, we get:

R = 2U
U = 0.67R

Normalizing so that R + U = 1, we get:

R = 0.4
U = 0.6

Therefore, the steady-state vector is [0.4, 0.6], which corresponds to choice D.

Suppose a Government Study Estimated That the Probability of Successive