Mathematics 452 [F01]: Midterm examination #1
Department of Mathematics and Statistics, University of Victoria
Oct. 22, 2010
This examination paper consists of 4 problems worth a total of 22 marks. Read through carefully and
choose the questions you want to answer. You need to total out 20 marks. However, feel free to answer as
many questions as you want. Any points that exceed 20 will be counted as
bonus marks
, i.e, the exam is
on 20 marks plus 2
bonus
points.
1. Consider a Markov chain on a countable state space with a probability transition matrix
P
= [
p
ij
].
Let
i
and
j
be two states such that
i
is not accessible from
j
.
Show that if
P
ij
>
0, then
i
is
transient.
[4]
2. Consider the Markov chain
X
n
whose transition probability matrix is given by
[8]
P
=
1
/
3
2
/
3
0
0
0
0
4
/
5
1
/
5
1
0
0
0
0
0
0
1
.
(a) Draw the graph or diagram of this Markov chain and use this graph to classify its sates into
recurrent and transient classes. Let
T
be the set of all transient states.
[2]
(b) Let
i, j
be two transient states (i.e
i, j
∈
T
) and
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 Spring '11
 stephenlang
 Calculus, Statistics, Probability, Markov chain, Andrey Markov

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