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IOE 316 – Homework 1 Solutions
Due October 30, 2007
1. (20 points) Three white and three black balls are distributed in two urns in such a way that
each contains three balls. We say that the system is in state
i,i
= 0
,
1
,
2
,
3, if the ﬁrst urn
contains
i
white balls. At each step, we draw one ball from each urn and place the ball drawn
from the ﬁrst urn into the second, and conversely with the ball from the second urn. Let
X
n
denote the state of the system after the
n
th
step. Explain why
{
X
n
,n
= 0
,
1
,
2
,...
}
is a
Markov chain and calculate its transition probability matrix.
SOLUTION:
(10 points for justiﬁcation of Markov chain, 10 points for correct t.p.m.)
X
n
is a Markov chain because the number of balls that will be in either urn in the future is
only dependent on the number of balls presently in the urns. There are four states
{
0
,
1
,
2
,
3
}
which represent the number of white balls that are in the ﬁrst urn.
The transition probability matrix looks as follows:
P
=
0
1
0
0
1
9
4
9
4
9
0
0
4
9
4
9
1
9
0
0
1
0
2. (20 points) Suppose that whether or not it rains today depends on previous weather conditions
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This homework help was uploaded on 04/02/2008 for the course IOE 316 taught by Professor Dolinskaya during the Spring '08 term at University of Michigan.
 Spring '08
 Dolinskaya

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