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Compute the probability that, in the next ten years, it is strong at least two years

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Test of Discrete Event Systems - 03.12.2014

Exercise 1

A study of the strengths of the football teams of the main American universities shows that, if a university has a strong team one year, it is equally likely to have a strong team or an average team next year; if it has an average team, it is average next year with 50% probability, and if it changes, it is just as likely to become strong as weak; if it is weak, it has 65% probability of remaining so, otherwise it becomes average.

1. What is the probability that a team is strong on the long run?

2. Assume that a team is weak. How many years are needed on average for it to become strong?

3. Assume that a team is weak. Compute the probability that, in the next ten years, it is strong at least two years.

Exercise 2

Consider the discrete-time homogeneous Markov chain whose state transition diagram is represented in the figure, and with transition probabilities p0

,1 = 1/3, p1

,2 = 1/8, p1

,3= 1/4 and p2

,3 = 4/5.

0

1

2

3 4

1. Compute the average recurrence time for each recurrent state.

2. Compute the stationary probabilities of all states, assuming the probability vector of the initial state π(0) = [ 1 0 0 0 0 ].

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