Math331, Spring 2008
Instructor: David Anderson
2nd Markov Chain Homework
1. Consider a Markov chain transition matrix
P
=
1
/
2
1
/
3
1
/
6
3
/
4
0
1
/
4
0
1
0
.
(a) Show the this is a regular Markov chain.
(b) If the process is started in state 1, find the probability that it is in state 3 after
two steps.
(c) Find the limiting probability vector
w
.
(d) Find lim
n
→∞
p
n
(
x,y
) for all
x,y
∈
S
. Why do you know these limits exist?
Solution:
(a) We have that
P
3
=
1
/
2
1
/
3
1
/
6
9
16
1
/
4
3
/
16
3
/
8
1
/
2
1
/
8
and so the Markov chain is regular.
(b) We have
P
2
=
1
/
2
1
/
3
1
/
6
3
/
8
1
/
2
1
/
8
3
/
4
0
1
/
4
.
Thus
P
(
X
2
= 3

X
0
= 1) =
P
2
(1
,
3) = 1
/
6.
(c) We have to solve the system of equations
xP
=
x
=
⇒
[
x
1
,x
2
,x
3
]
1
/
2
1
/
3
1
/
6
3
/
4
0
1
/
4
0
1
0
= [
x
1
,x
2
,x
3
]
.
Subject to
x
1
+
x
2
+
x
3
= 1.
Therefore, we must solve the linear system of
equations
(1
/
2)
x
1
+ (3
/
4)
x
2
=
x
1
(1
/
3)
x
1
+
x
3
=
x
2
(1
/
6)
x
1
+ (1
/
4)
x
2
=
x
3
x
1
+
x
2
+
x
3
= 1
.
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 Spring '08
 Anderson
 Math, Probability, Markov chain, lim Pn, Markov Chain Homework

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