\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
A computer server is checked every hour. Its status can be Operational (O) or Under Maintenance (M).
Studies show that:
  • If the server is Operational, there is a \(10\pourcent\) chance it will be Under Maintenance by the next hour.
  • If the server is Under Maintenance, there is a \(30\pourcent\) chance it will become Operational again by the next hour.
Let \(X_n\) be the state of the server at hour \(n\).
  1. Justify that the sequence \((X_n)\) forms a time-homogeneous Markov chain.
  2. Identify the state space \(E\) and determine the transition probabilities \(p_{ij}\) for \(i, j \in E\).

Capture an image of your work. AI teacher feedback takes approximately 10 seconds.