STA 3007 Applied Probability
2005
Tutorial 5
1. The Long Run Behavior of Markov Chains
Exercise:
i. Suppose that a production process changes state according to a Markov process whose
transition probability matrix is given by:
P
=
0
1
2
3
0
0
.
3
0
.
5
0
.
0
0
.
2
1
0
.
5
0
.
2
0
.
2
0
.
1
2
0
.
2
0
.
3
0
.
4
0
.
1
3
0
.
1
0
.
2
0
.
4
0
.
3
It is known that
π
1
=
119
379
= 0
.
3140
and
π
2
=
81
379
= 0
.
2137
a Determine the limiting probabilities
π
0
and
π
3
.
b Suppose that states 0 and 1 are ”InControl” while states 2 and 3 are deemed ”Out
of Control”. In the long run, what fraction of time is the process Out of Control?
c In the long run, what fraction of transitions are from an InControl state to an Out
ofControl state?
1
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ii. Suppose that a production process changes state according to a Markov process whose
transition probability matrix is given by:
P
=
0
1
2
3
0
0
.
2
0
.
2
0
.
4
0
.
2
1
0
.
5
0
.
2
0
.
2
0
.
1
2
0
.
2
0
.
3
0
.
4
0
.
1
3
0
.
1
0
.
2
0
.
4
0
.
3
a Determine the limiting distribution for the process.
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 Spring '11
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 Markov Chains, Probability, Markov chain, Andrey Markov, two days, transition probability matrix

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