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A company has two machines. During any day, each machine that is working at the beginning of the day has a 1/3 chance of breaking down. If a machine breaks down during the day, it is sent to a repair facility and will be working two days after it breaks down. (Thus, if a machine breaks down during day 3, it will be working at the beginning of day 5.) Letting the state of the system be the number of machines working at the beginning of the day, formulate a transition probability matrix for this situation. Suppose that tomorrow’s Smalltown weather depends on the last two days of Smalltown weather, as follows: (1) If the last two days have been sunny, then 95% of the time, tomorrow will be sunny. (2) If yesterday was cloudy and today is sunny, then 70% of the time, tomorrow will be sunny. (3) If yesterday was sunny and today is cloudy, then 60% of the time, tomorrow will be cloudy. (4) If the last two days have been cloudy, then 80% of the time, tomorrow will be cloudy. Using this information, model Smalltown’s weather as a Markov chain. If tomorrow’s weather depended on the last three days of Smalltown weather, how many states will be needed to model Smalltown’s weather as a Markov chain? (Note: The approach used in this problem can be used to model a discrete-time stochastic process as a Markov chain even if X_{t+1} depends on states prior to X_{t}, such as X_{t-1} in the current example.)
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