Math 152, Spring 2010
Assignment #8
Notes:
•
Each question is worth 5 marks.
•
Due in class: Monday, March 15 for MWF sections; Tuesday, March 16
for TTh sections.
•
Solutions will be posted Tuesday, March 16 in the afternoon.
•
No late assignments will be accepted.
1. Consider a random walk with
n
states (this number
n
will be known for
a given case, but we are also interested in considering it for general
n
). It
has the following properties:
•
If it is in states 1 to
n
−
1 (that is, any state but the last) is is equally
likely to stay in that state or move to the state numbered one higher.
•
If it is in the last state (number
n
) then it remains there forever.
(a)
Write down the transition matrix
P
for the case of three states (that
is,
n
= 3).
(b)
For the system above with three states, if it starts in state 1, what is
the chance it will be in state 3 after 3 time steps?
(c)
Write lines of MATLAB commands that would generate the matrix
P
for general
n
.
Hint:
use a
for
loop.
Solution
:
a) Suppose the particle starts in state 1. Then there is a chance of 1
/
2
that it stays in state 1, and a 1
/
2 chance that it moves to state 2. It never
moves to state 3. The vector of probabilities is (1
/
2
,
1
/
2
,
0).
If the particle starts in state 2, it has 1
/
2 chance of staying and 1
/
2
chance of moving to state 3. It never goes to state 1. Thus the vector of
probabilities is (0
,
1
/
2
,
1
/
2).
Finally, if the particle starts in state 3, it never leaves, so the vector of
probabilities is (0
,
0
,
1).
Thus the matrix
P
describing the walk is
1
/
2
0
0
1
/
2
1
/
2
0
0
1
/
2
1
.
b) There are di
ff
erent ways to do this problem. The easiest would form
the initial vector
e
=
1
0
0
, which represents a particle in state 1. Then
the passage of each time step corresponds to multiplying by
P
, and we
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have to do this three times.
We can multiply out successively to get
Pe, P
2
e
= (
P
(
Pe
), and
P
3
e
=
P
(
P
2
e
), and we find that
f
=
P
3
e
=
1
/
8
3
/
4
1
/
2
,
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 Spring '08
 Caddmen
 Math, Linear Algebra, Invertible matrix

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