ORIE 361/523 – Homework 4
Instructor: Bikramjit Das
due July 27, 2007
Answers.
1. If
Σ
i
P
i,j
= 1
∀
j,
then,
r
j
=
1
M
+1
satisﬁes
r
j
= Σ
m
i
=0
r
i
P
i,j
1 = Σ
m
i
=0
r
i
Hence by uniqueness of stationary distribution(since the chain is irreducible and aperiodic),
these are the limiting probabilities.
2. Consider the Markov chain which has 2 states, S if the dividend has been suspended and N
if it is not.
Let,
X
n
= The status of dividend payment on the nth observed period.
Then, the transition matrix P is
P
=
S
N
±
0
.
6 0
.
4
0
.
1 0
.
9
²
This is clearly an irreducible Markov chain, which has ﬁnite state space. So, we know, the
long run proportions of staying in state j is
π
j
, where
π
is the stationary distribution for the
Markov chain. We proceed to compute the stationary distribution.
πP
=
π
X
i
π
i
= 1
gives
0
.
6
π
S
+ 0
.
1
π
N
=
π
S
0
.
4
π
S
+ 0
.
9
π
N
=
π
N
π
S
+
π
N
= 1
Solving, we get
π
N
=
4
5
, which is the required answer.
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 Summer '07
 SHMOYS/LEWIS
 Markov chain, stationary distribution, Bikramjit Das

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