final 307 2017.pdf - Math 307 Sec. 202 (TZOU) FINAL April...

Math 307 – Sec. 202 (TZOU) FINAL — April 18, 2017 Page 1 of 15
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This exam has 7 questions on 15 pages , for a total of 84 points.
Duration: 2.5 hours
• Write your name or your student number on every page.
• You need to show enough work to justify your answers.
• Continue on the back of the page if you run out of space.
• You have to turn in all 15 pages of this booklet even if you don’t use all the pages.
• This is a closed-book examination. None of the following are allowed : documents
or electronic devices of any kind (including calculators, cell phones, etc.)
Question: 1 2 3 4 5 6 7 Total
Points: 10 6 14 21 15 10 8 84
Score:

Math 307 – Sec. 202 (TZOU) FINAL — April 18, 2017 Page 2 of 15
1. In a warm climate where it never snows, the weather on any particular day is either
sunny, cloudy, or rainy (assume that the weather is constant for the entire day). If it is
sunny, there is a 1 / 2 chance that it will remain sunny the next day, 1 / 3 chance that it
will be cloudy the next day, and 1 / 6 chance that it will rain the next day. If it is cloudy,
the next day will be sunny, cloudy, or rainy with equal probability. If it is raining, there
is a 1 / 6 chance that it is sunny the next day, a 1 / 3 chance that it is cloudy the next
day, and a 1 / 2 chance that it remains rainy the next day. Denote the probability of sun,
cloud, and rain on the k -th day by s
k
, c
k
, and r
k
, respectively. Denote also the vector
v
k
= ( s
k
, c
k
, r
k
)
T
.
(a) 2 marks Show that λ = 1 must be an eigenvalue of any n × n matrix M whose columns each
sum to one. That is, show that if
∑
n
i =1
M
ij
= 1 for j = 1 , . . . , n , then λ = 1 is an
eigenvalue of M .
(b) 3 marks Construct the transition matrix A such that v
k +1
= A v
k
(you will receive 1 point
for each correct column). Here, v
k
is defined above.
(c) 2 marks The eigenpairs of A are such that A x
1
= λ
1
x
1
, A x
2
= λ
2
x
2
, and A x
3
= λ
3
x
3
, where
λ
1
= 1, and | λ
2
| , | λ
3
| < 1. Suppose the eigenvectors x
1
, x
2
, and x
3
are mutually
orthogonal (but not necessarily orthonormal). For arbitrary initial distribution v
0
,
find the coefficients d
1
, d
2
, and d
3
such that v
0
= d
1
x
1
+ d
2
x
2
+ d
3
x
3
. Do not
leave your answer in terms of the inverse of a certain matrix. Do not compute the
eigenvectors.
(d) 2 marks Use the result of Part (c) to express v
20
in terms of x
1
, x
2
, and x
3
, along with the
coefficients d
1
, d
2
, and d
3
. Do not compute the eigenvectors.
(e) 1 mark Express the steady state distribution v
∞
in terms of x
1
and/or x
2
and/or x
3
, along
with the coefficients d
1
and/or d
2
and/or d
3
. Do not compute the eigenvectors.

Math 307 – Sec. 202 (TZOU) FINAL — April 18, 2017 Page 3 of 15
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