ORIE 361/523 – Homework 4
Instructor: Mark E. Lewis
due February 20, 2008 (drop box)
1.
X
n
is a Markov Chain on state space
{
1
,
2
,
3
}
with transition probability matrix
P
=
0
1
0
0
1
/
4 3
/
4
2
/
5 3
/
5
0
.
Estimate
P
(
X
n
= 1

X
0
= 1) for large
n
.
2. In unproﬁtable times, corporations sometimes suspend dividend payments. Suppose that
after a dividend has been paid, the next one will be paid with probability 0.9, while after a
dividend is suspended the next one will be suspended with probability 0.6. In the long run,
what is the fraction of dividends that will be paid?
3. A certain town never has two sunny days in a row. Each day is either classiﬁed as being
either sunny, cloudy (but dry), or rainy. If it is sunny one day, then it is equally likely to be
either cloudy or rainy the next day. If it is rainy or cloudy one day, then with probability 1/2
it will be the same the next day, and if it changes it is equally likely to be either of the two
possibilities. In the long run, what proportion of days are sunny? What proportion of days
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 Spring '07
 LEWIS,M.
 Markov chain, one day, Mark E. Lewis

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