m358k-hw-six-solns
pdf
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School
University of Texas *
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Course
358K
Subject
Industrial Engineering
Date
Dec 6, 2023
Type
Pages
5
Uploaded by ConstableGiraffeMaster829
Homework assignment #6: Solutions
Milica Cudina
2021-10-08
Textbook exercises
Problem 1. (
1 + 1 + 1 + 2 = 5
points)
Solve
Problem 4.2
from the textbook.
Solution:
knitr
::
include_graphics
(
"oc-p4-2.png"
)
Problem 2. (
1 + 3 + 2 + 2 + 2 + 3 = 13
points)
Solve
Problem 4.4
from the textbook.
Solution:
knitr
::
include_graphics
(
"oc-p4-4.png"
)
1
Problem 3. (
3 + 3 = 6
points)
Solve
Problem 4.6
from the textbook.
Solution:
knitr
::
include_graphics
(
"oc-p4-6.png"
)
2
Problem 4. (
3 + 3 = 6
points)
Solve
Problem 4.8
from the textbook.
Solution:
knitr
::
include_graphics
(
"oc-p4-8.png"
)
Problem 5. (
5
points)
Solve
Problem 4.10
from the textbook.
Solution:
knitr
::
include_graphics
(
"oc-p4-10.png"
)
3
Your preview ends here
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Additional problems
Problem 6. (
3
×
2 = 6
points)
Let
Z
be a standard normal random variable.
Using the standard normal tables, calculate the following
probabilities:
(i)
P
[
-
1
.
23
< Z <
2
.
37]
(ii)
P
[1
/Z <
1]
(iii)
P
[
Z
2
>
2
.
56]
Solution:
(i)
P
[
-
1
.
23
< Z <
2
.
37] =
P
[
Z <
2
.
37]
-
P
[
Z
≤ -
1
.
23]
pnorm
(
2.37
)
-
pnorm
(
-
1.23
)
## [1] 0.8817574
(ii)
P
[1
/Z <
1] =
P
[
Z <
0] +
P
[
Z >
1] =
P
[
Z <
0] + 1
-
P
[
Z
≤
1]
pnorm
(
0
)
+
1
-
pnorm
(
1
)
## [1] 0.6586553
(iii)
P
[
Z
2
>
2
.
56] =
P
[
|
Z
|
>
1
.
6] =
P
[
Z <
-
1
.
6] +
P
[
Z >
1
.
6] = 2
P
[
Z <
-
1
.
6]
2
*
pnorm
(
-
1.6
)
## [1] 0.1095986
Problem 7. (
4 + 5 = 9
points)
Source: Problem #139 from Moore-McCabe-Craig.
The interquartile range (IQR) of a distribution is defined as the distance between the first and the third
quartiles.
(i) (
4
points) What is the IQR for the standard normal distribution?
(ii) (
5
points) What is the IQR for a normal distribution with mean
μ
and variance
σ
2
?
Solution:
(i) The value
z
*
of the third quartile can be obtained as
z
*
= Φ
-
1
(0
.
75)
qnorm
(
0.75
)
## [1] 0.6744898
By the symmetry of the standard normal distribution, we have
-
z
*
= Φ
-
1
(0
.
25)
4
qnorm
(
0.25
)
## [1] -0.6744898
Therefore the IQR for the standard normal distribution is
qnorm
(
0.75
)
-
qnorm
(
0.25
)
## [1] 1.34898
(ii) Any normal random variable
X
∼
Normal
(
mean
=
μ, variance
=
σ
2
)
can be represented as a linear
transformation of the standard normal random variable
Z
. Namely, we have
X
=
μ
+
σZ.
So, the interquartile range is about
1
.
34898
σ
.
5