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ST3236: Stochastic Process Tutorial 6

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Presentation on theme: "ST3236: Stochastic Process Tutorial 6"— Presentation transcript:

1 ST3236: Stochastic Process Tutorial 6
TA: Mar Choong Hock Exercises: 7

2 Question 1 Consider the MC with transition probability matrix
Determine the limiting distribution.

3 Question 1 Let  = (0, 1, 2) be the limiting distribution, we have
deleting one of the first three equations, we have the solution as 0 = , 1 = , 2 =

4 Question 2 Consider the MC with transition probability matrix
What fraction of time, in the long run, does the process spend in state 1?

5 Question 2 Let  = (0, 1, 2) be the limiting distribution, we have
deleting one of the first three equations, we have the solution as 0 = , 1 = , 2 =

6 Question 2 With frequency 1 = , in the long run, does the process spend in state 1

7 Question 3 Consider the MC with transition probability matrix
Every period that the process spends in state 0 incurs a cost of 2$. Every period that the process spends in state 1 incurs a cost of 5$. Every period that the process spends in state 2 incurs a cost of 3$. What is the long run average cost per period associated with this Markov chain.

8 Question 3 Let  = (0, 1, 2) be the limiting distribution, we have
deleting one of the first three equations, we have the solution as 0 = , 1 = , 2 =

9 Question 3 The long run average cost per period associated with this Markov chain is x x x 3 = $

10 Question 4 Suppose that the social classes of successive
generations in a family follows a Markov chain with transition probability matrix given by What fraction of families are upper class in the long run?

11 Question 4 Let  = (L, M, U) be the limiting distribution, we have
deleting one of the first three equations, we have the solution as L = , M = , U =

12 Question 4 The fraction of families are upper class in the long run is U =


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