ORIE 361/523 : Introduction to Stochastic Processes
Summer2007
Problem Set 4
Due in one week, on July 27 in class
.
1. (10 points, page 266, problem 20) A transition matrix
P
is said to be
doubly stochastic
if the
sum over each column equals one; that is,
X
i
P
ij
= 1
,
for all j
.
If such a chain is irreducible and aperiodic and consists of
M
+ 1 states 0
,
1
, . . . , M
, show
that the limiting probabilities are given by
π
j
=
1
M
+ 1
,
j
= 0
,
1
, . . . , M
2. (10 points) In unprofitable times, corporations sometimes suspend dividend payments. Sup
pose 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. (15 points)
Harry and the comely young lady.
Harry, the famous restaurateur, has his
eye on a comely young lady whom he originally spotted at a high cholesterol cooking course.
However, during the course Harry never had the courage to introduce himself as a famous
restaurateur. The young lady eats lunch regularly on Optima Street where Harry’s restaurant
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 Summer '07
 SHMOYS/LEWIS
 Probability, Probability theory, Stochastic process, Markov chain

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