Statistics 348, Introduction to Stochastic Processes, Winter 2011
Midterm Review Problems
1. First just a little note: remember the following basic calculus fact that
∞
X
n
=1
1
n
β
=
∞
,
β
≤
1
<
∞
,
β >
1
If you want to check it for yourself, do the integral
Z
∞
1
dx
x
β
and note that it depends on
β
in the same way. The integral and the sum are pretty much the same thing, so
you can use one to check how the other behaves.
2.
Simple Random Walk with Absorbing Barriers.
Consider simple random walk on
{
0
,
1
,
2
, . . . , N
}
with
absorbing barriers at 0 and
N
. Assume there is a probability
p
chance of going up and a
q
= 1

p
chance of
going down.
(a) Let
p
k
=
P
(hit N before 0

X
0
=
k
).
Derive a difference equation for the
p
k
.
What are the boundary
conditions?
(b) Solve the above difference equation. Remember that the
p
= 1
/
2 case is different from the
p
6
= 1
/
2 case.
(c) Let
D
k
=
E
[time until absorption at 0 or N

X
0
=
k
]. Derive a difference equation for the
D
k
. What are
the boundary conditions?
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 Spring '11
 nomane
 Statistics, Markov Chains, Stochastic process, Markov chain, Random walk, simple random walk

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