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A software company tracks its users monthly. A user can be in one of three states:
  • Premium (P): The user pays for the full service.
  • Free (F): The user uses the limited free version.
  • Inactive (I): The user has cancelled their account.
The marketing department observes the following monthly trends:
  • Of the Premium users, \(80\pourcent\) keep their subscription, \(15\pourcent\) switch to the Free version, and \(5\pourcent\) cancel.
  • Of the Free users, \(20\pourcent\) upgrade to Premium, \(70\pourcent\) stay Free, and \(10\pourcent\) cancel.
  • Of the Inactive users, \(10\pourcent\) reactivate with a Premium subscription, \(10\pourcent\) reactivate with a Free account, and \(80\pourcent\) remain inactive.
Let \(X_n\) be the random variable representing the state of a randomly chosen user in month \(n\).
  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 \(i, j \in E\).

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