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IEOR 3106 Solutions to Midterm Exam I
1. (40 points) A small business sells professional digital cameras one by one. On
any given day, either a camera is delivered to them from the factory (causing the
inventory level to go up by 1) with probability
p
=0
.
20, or they sell a camera
(causing the inventory level to go down by 1), with probability 1
−
p
.
80.
(a) (10 points) Suppose that there are initially 3 cameras in inventory. What is
the probability that the inventory level drops to 0 before reaching 5?
SOLUTION:
Gambler’s ruin problem with
N
=5
,
i
=3
,
p
.
20;
q/p
=4
.
We want 1
−
P
3
, where
P
3
=
1
−
(
)
3
1
−
(
)
5
=63
/
1023 = 0
.
062
(b) (10 points) Suppose that there are initially 3 cameras in inventory. What is
the probability that the inventory level reaches 4 before going down to 1?
SOLUTION:
Gambler’s ruin problem (
a
=1
,b
=2
) with
N
,
i
,
p
.
20. We want
P
2
=
1
−
(
)
2
1
−
(
)
3
=15
/
63=0
.
24
For the next three questions, suppose further that if the inventory
level ever reaches the high value of
5
, then they sell
4
of them at once
(same day) to a wholesaler at a reduced rate. If the inventory level
ever drops to
0
, then
2
cameras are immediately delivered (same
day).
(c) (10 points) Find the longrun proportion of days that the inventory level hits
0.
SOLUTION:
Model as a MC on
S
=
{
1
,
2
,
3
,
4
}
:
X
n
=inventory level at
the end of the
n
th
day. We then will solve for
π
=
πP
yielding our answer as
(1
−
p
)
π
1
, since the level hits 0 if and only if it does so by reaching level 1 and
then selling a camera that day. For our chain however,
P
1
,
2
=1 since when
X
n
=1 either
X
n
+1
=2 due to 1 camera arriving (prob.=
p
), or
X
n
+1
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 Spring '08
 Stochastic

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