STAT 333 Assignment 3
Due: Tuesday, July 27 at the beginning of class
1.
Consider a discretetime Markov Chain with state space S = {0, …, 7} and transition matrix
P
=
⎝
⎜
⎜
⎜
⎜
⎛
1/2
1/4
0
0
0
0
1/4
0
1/4
0
0
0
0
0
3/4
0
0
0
1/3
0
0
0
0
2/3
0
1/5
1/5
1/5
1/5
1/5
0
0
1/6
0
0
0
1/3
1/6
1/6
1/6
0
0
0
0
0
1
0
0
1/4
0
0
0
0
0
3/4
0
0
0
2/3
0
0
0
0
1/3
⎠
⎟
⎟
⎟
⎟
⎞
a.
Determine the classes of this chain, which are open or closed, write
P
in simplified form,
and find the period of each closed class.
b.
Find the equilibrium distribution corresponding to each closed class. Write down the
general form of all stationary distributions for this chain.
c.
Find the absorption probabilities from each transient state into each closed class.
d.
Describe the longrun behaviour of the chain if X
0
= 0. Do the same if X
0
= 3.
2.
NOTE: You may use mathematical software for this question, just include your output.
Consider a discretetime Markov Chain with five transient states (1, 2, 3, 4, and 5) and two
recurrent states (6 and 7). The chain is equally likely to start in any of the transient
P
=
⎝
⎜
⎜
⎜
⎛
1/2
0
1/4
0
0
0
1/4
0
1/3
0
1/3
0
1/3
0
1/5
1/5
1/5
1/5
1/5
0
0
0
0
0
1/4
1/4
1/4
1/4
2/3
0
1/3
0
0
0
0
0
0
0
0
0
1/2
1/2
0
0
0
0
0
1/3
2/3
⎠
⎟
⎟
⎟
⎞
states.
The transition matrix is
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 Spring '08
 Chisholm
 Probability theory, Poisson process, Markov chain, discretetime Markov chain

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