Practice Midterm 1_annotated
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Gwinnett Technical College *
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Course
3339
Subject
Industrial Engineering
Date
Feb 20, 2024
Type
Pages
24
Uploaded by BarristerStrawElk2529
ISyE 3232
Stochastic Manufacturing and Service Models
Spring 2024
Practice Midterm 1
Jan. 30, 2024
I,
, do swear that I abide by the Georgia Tech Honor Code. I understand that any honor code
violations will result in a failure (an F).
Signature:
Date:
You are permitted to have one one-sided and handwritten cheat sheet on a 8.5”
×
11” or smaller sheet of paper.
You may use a calculator, although the exam was designed so that one is not necessary.
Some formulas you may need:
∑
N
i
=0
=
N
(
N
+1)
2
for any integer
N
.
Uniform distribution (continuous):
Let
X
∼
Uniform(
a, b
). Then,
f
(
x
) =
1
b
-
a
for
a
≤
x
≤
b
and
f
(
x
) = 0 otherwise.
E
[
X
] = 1
/
2(
a
+
b
),
Var
(
X
) = 1
/
12(
b
-
a
)
2
.
F
(
x
) =
x
-
a
b
-
a
for
a
≤
x
≤
b
,
F
(
x
) = 0 for
x
≤
a
and
F
(
x
) = 1 for
x
≥
b
.
Poisson distribution:
Let
X
∼
Poisson(
λ
). Then,
f
(
x
) =
λ
x
e
-
λ
x
!
for any
x
∈ {
0
,
1
,
2
, ...
}
and
f
(
x
) = 0 otherwise.
E
[
X
] =
Var
(
X
) =
λ
.
F
(
x
) =
e
-
λ
∑
x
i
=0
λ
i
i
!
Geometric distribution:
Let
X
∼
Geometric(
p
). Then,
f
(
x
) = (1
-
p
)
x
-
1
p
for any
x
∈ {
0
,
1
,
2
, ...
}
and
f
(
x
) = 0 otherwise.
E
[
X
] = 1
/p
,
Var
(
X
) = (1
-
p
)
/
(
p
2
).
F
(
x
) = 1
-
(1
-
p
)
x
Exponential distribution:
Let
X
∼
exponential(
λ
). Then,
f
(
x
) =
λe
-
λx
for any
x
≥
0 and
f
(
x
) = 0 otherwise.
E
[
X
] = 1
/λ
.
Var
(
X
) = 1
/λ
2
.
F
(
x
) = 1
-
e
-
λx
.
2
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Multiple choice (30 points):
For each of the following questions, please select one answer (3 points each).
If you select more than one, the question will
automatically be marked wrong.
1. Let
X
be a continuous random variable with pdf
f
(
x
). What is
E
[(5
-
X
)
+
]?
(a)
R
∞
0
min
{
5
-
x,
0
}
f
(
x
)
dx
(b)
R
∞
5
min
{
5
-
x,
0
}
f
(
x
)
dx
(c)
R
5
0
max
{
5
-
x,
0
}
f
(
x
)
dx
(d)
R
5
-∞
max
{
5
-
x,
0
}
f
(
x
)
dx
2. Let
Var
(
X
) denote the variance of
X
. Which of the following is FALSE?
(a)
Var
(
aX
) =
|
a
|
Var
(
X
) for any constant
a
.
(b)
Var
(
X
) =
E
[(
X
-
E
[
X
])
2
].
(c)
Var
(
X
+
Y
) =
Var
(
X
) +
Var
(
Y
) for any two independent random variables
X
and
Y
.
(d)
Var
(
X
+
a
) =
Var
(
X
) for any constant
a
.
3. Let
X
be a Poisson random variable with rate parameter
λ
= 6. What is
E
[
X
2
]?
(a) 6
(b) 36
(c) 42
(d) 0
4. Consider a newsvendor problem where demand follows a discrete probability distribution. In a cost-minimization setting, the optimal production quantity:
(a) is the smallest
y
such that
F
(
y
)
≥
c
u
+
c
v
c
u
+
c
o
.
(b) is any
y
such that
F
(
y
)
>
c
u
-
c
v
c
u
+
c
o
.
(c) only exists if there is a
y
such that
F
(
y
) =
c
u
-
c
v
c
u
+
c
o
.
(d) the smallest
y
such that
F
(
y
)
≥
c
u
-
c
v
c
u
+
c
o
.
5. For a continuous random variable
X
with pdf
f
(
x
), which of the following is FALSE?
(a)
f
(
x
)
≥
0 for all
x
.
(b)
f
(
x
)
≤
1 for all
x
.
(c)
E
[
X
] =
R
∞
-∞
xf
(
x
)
dx
.
(d) The CDF is defined by
F
(
x
) =
R
x
-∞
f
(
x
)
dx
.
3
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6. Let
E
[
X
] denote the expected value of a random variable
X
. Which of the following is false?
(a)
E
[
cX
] =
c
E
[
X
] for a constant
c
.
(b)
E
[
X
+
Y
] =
E
[
X
] +
E
[
Y
] for a random variable
Y
.
(c)
E
[
X
2
] =
E
[
X
]
2
.
(d)
E
[
X
] =
∑
K
k
=1
E
[
X
|
S
k
]
P
(
S
k
) for any partition of the sample space
S
1
, ..., S
K
.
7. Let
X
be a geometric distribution with success probability
p
= 0
.
25. What is
E
[(3
-
X
)
+
]? Use the pdf given on the cover page.
(a) 1.3275
(b) 1.6875
(c) -1.6875
(d) 0
8. Suppose that
E
[
X
] = 4 and
Var
(
X
) = 10. What is
E
[3
X
2
+ 4]?
(a) 22
(b) 52
(c) 148
(d) 82
9. Let
y
*
denote the optimal order quantity for a profit maximization problem where the demand
D
has a discrete distribution. Let
p
be the selling price,
b
be the backorder cost,
c
v
be the buying price, and
h
be the holding cost. Which of the following must be true?
(a)
F
(
y
*
) =
p
+
b
-
c
v
p
+
b
+
h
, where
(b) The order quantity
y
*
minimizes cost.
(c)
F
(
y
*
)
<
p
+
b
-
c
v
p
+
b
+
h
.
(d) The order quantity
y
*
maximizes long-run average profit.
10. Suppose that the time it takes to serve a customer at Rise and Roll is exponentially distributed with a mean of 5 minutes. What is the probability that it
takes longer than 5 minutes for you to be served?
(a) 0.5
(b) 0.6321
(c) 0
(d) 0.3679
Short Answer:
For the remaining problems, you must show all of your work to receive full credit.
Simply showing the final answer without any work will result in 0 points, even if the final answer is correct.
If you need to use an answer from a previous part, and you have not obtained a solution for it, use the capital letter corresponding to that part.
For
example, plug in
A
anywhere where you need to use the solution from part (a).
If you run out of space, please use the back of the page and indicate that you have done so.
4
11. (20 points) Polymerase chain reaction (PCR) tests are often used to determine if someone has COVID-19 or not. PCR test will return either a positive
or negative result. However, PCR tests may sometimes provide imperfect results. That is, the PCR test may give a positive result given that the patient
does not actually have COVID-19 (i.e., false-positive) or a negative result given that the patient has COVID-19 (i.e., a false-negative). For PCR tests, the
false-positive probability is given by
p
F P
= 0
.
01 and the false-negative probability is
p
F N
= 0
.
10. Additionally, suppose that 15% of patients who get tested
for COVID-19 actually have COVID-19.
(a) (5 points) The
test-sensitivity
, which is also called the true-positive probability
p
T P
, is equal to the probability of getting a positive result given that
the patient has COVID-19. Likewise, the
test-specificity
, which is also called the true-negative probability,
p
T N
, is equal to the probability of getting
a negative result given that the patient does not have COVID-19. What are the values of
p
T P
and
p
T N
?
5
(b) (5 points) Accuracy refers to the expected proportion of tests that give a correct result. What is the accuracy of PCR tests?
6
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(c) (5 points) At-home COVID tests are cheaper to manufacture, but may not detect COVID-19 as well as PCR tests do.
However, they are slightly
better at ruling out COVID-19. For at-home tests, the false-positive probability is
q
F P
= 0
.
005 and the false-negative probability if
q
F N
= 0
.
30. What
is the accuracy of this test?
7
(d) (5 points) What proportion of patients who get tested for COVID-19 need to be positive with COVID-19 so that at-home tests are more accurate
than PCR tests?
8
12. (35 points) After botching the sale of Taylor Swift’s concert tickets, Ticketmaster was required to wholesale a significant proportion of its tickets for the
next Taylor Swift concert. In particular, they must sell tickets for
20 each to their competitors. Due to federal regulations on price gouging, competitors
can only sell tickets for
60 each on the day before the concert, but cannot sell tickets on the day of the concert. If competitors do not have enough tickets
to meet demand, unmet demand is lost. Leftover tickets on the day of the concert are then sold to scalpers at
10 each. Suppose that demand for tickets
on the day before the concert can be modeled as a discrete random variable with the following pmf:
f
(
x
) =
1
/c
x
∈ {
0
,
1
,
2
,
3
, ...,
59
}
2
/c
x
∈ {
60
,
61
,
62
, ...,
89
}
3
/c
x
∈ {
90
,
91
, ...,
100
}
,
where
c
is a constant. Suppose you are one of Ticketmaster’s competitors.
(a) (3 points) What is the value of the constant
c
?
9
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(b) (7 points) What is your expected profit if you order 60 tickets?
10
(c) (10 points) How many tickets should be purchased from Ticketmaster to maximize expected profit?
11
(d) (3 points) Suppose now that the random demand follows a discrete uniform distribution with pmf
f
2
(
x
) =
(
1
k
x
∈ {
0
,
1
, ...,
100
}
0
otherwise,
where
k
is a constant. What is the value of
k
?
12
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(e) (10 points) Compute the new optimal order quantity under demand pmf
f
2
(
x
).
13
(f) (2 point) Did the new optimal order quantity in part (e) increase or decrease compared to the previous optimal order quantity in part (c)? Based on
how the distributions are different, explain your intuition for why we might see this change in the optimal order quantity.
14
13. (15 points) A small Coca-Cola factory servicing Georgia Tech must decide how much Coke Zero to produce each day. The cost of producing Coke Zero
includes a
100 fixed cost associated with changing production lines, as well as a unit cost of
.5 per liter of Coke Zero produced. Demand is modeled fairly
accurately using a triangular distribution with the following pdf
f
(
x
) =
2
x
5000
0
≤
x
≤
50
200
-
2
x
5000
50
≤
x
≤
100
0
otherwise,
and cdf:
F
(
x
) =
0
x
≤
0
x
2
5000
0
≤
x
≤
50
1
-
(100
-
x
)
2
5000
50
≤
x
≤
100
1
x
≥
100
.
Since Coca-Cola cares deeply about its loyal customer-base, the cost of unmet demand is estimated at
1.50 per liter. The cost of storing any leftover Coke
Zero is
1 per liter.
(a) (7 points) What is the optimal volume of Coke Zero to produce?
15
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(b) (8 points) Suppose that we had 20 liters of Coke Zero in storage. What is the expected cost incurred by not producing any Coke Zero and only using
the Coke Zero left in storage?
16
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