ECMT2160: Computational Assignment
Due: October 24, 11:59pm
This assessment task requires you to use Matlab. You should prepare your
submission as a Matlab Live Script file (i.e., a .mlx file). Submit your answers
through the Canvas course website. Your submission should include a mixture of
written responses formatted as text, blocks of Matlab code, andMatlab output,
including graphs. I expect you to alternate between text and code blocks as I have
done in the lecture notes. You should submit two versions of your answers: the
original .mlx file, and a version exported to .html.
To complete the assessment task you will need to download a .mat file from
Canvas, under Files > Matlab > AssessmentMatFiles. You should use the file
named xxx.mat, where xxx is your Student ID (as listed in Canvas). This file
contains a 999× 999 matrix with the name . You can load it intoMatlab using
the command load(“xxx.mat”), with the .mat file saved in the current
Matlab working directory.
You may work on this assessment individually, or in pairs. If you work in
pairs (which I strongly recommend), it is important that you clearly indicate the
Student ID of your partner in your submission. Your submission should not be
identical to your partner’s submission because you will each be using different
.mat files.
Answer all questions. The assignment is worth a total of 15 points towards
your final assessment. Points will be deducted for poor presentation, including:
excessive typos, poor written expression, poor organization, etcetera. Do not
print any long vectors of numbers in your response.
You can ask questions about this assessment on the Ed discussion forum, but
please do not answer questions asked by others. I will answer questions as I
deem appropriate.
You are free to use Google or AI to help you with this assessment.
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To solve this assignment you will need to work with four real-valued func-
tions of two real variables ( and ). Each function also depends on a scalar
parameter that is equal to or greater than one. The first function, denoted
(,), is defined by
(,) =
(
−65 log()
)−1 (
−32 log()
)−1
for all and strictly between 0 and 1. If either or is not strictly between
0 and 1 then we define (,) = 0. The second function, denoted ℎ (,), is
defined by
ℎ (,) =
(
−65 log()
)
+
(
−32 log()
)
for all and strictly between 0 and 1. If either or is not strictly between 0
and 1 then we defineℎ (,) = 0. The third function, denoted (,), is defined
by
(,) = exp
(
−ℎ (,)1/
)
for all and strictly between 0 and 1. If either or is not strictly between 0
and 1 then we define ℎ(,) = 0. The fourth function, denoted (,), is defined
by
(,) = 95 (,)ℎ (,)
−2+1/
[
ℎ (,)1/ + − 1
]
(,)
for all and strictly between 0 and 1. If either or is not strictly between 0
and 1 then we define (,) = 0.
Answer all of the following questions.
1. (1 point.) Calculate (without displaying your answer) the 19 × 19 matrix
for which the entry in row and column is equal to
2
(

20 ,

20
)
.
Using this matrix and the mesh command inMatlab, display a 3D graph
of the function 2(,) for and strictly between 0 and 1.
2
2. (1 point.) Calculate (without displaying your answer) the 19 × 19 matrix
for which the entry in row and column is equal to
ℎ2
(

20 ,

20
)
.
Using this matrix and the mesh command inMatlab, display a 3D graph
of the function ℎ2(,) for and strictly between 0 and 1.
3. (1 point.) Calculate (without displaying your answer) the 19 × 19 matrix
for which the entry in row and column is equal to
2
(

20 ,

20
)
.
Using this matrix and the mesh command inMatlab, display a 3D graph
of the function 2(,) for and strictly between 0 and 1.
4. (1 point.) Calculate (without displaying your answer) the 19 × 19 matrix
for which the entry in row and column is equal to
2
(

20 ,

20
)
.
Using this matrix and the mesh command inMatlab, display a 3D graph
of the function 2(,) for and strictly between 0 and 1.
5. (1 point.) Calculate (without displaying your answer) the 999× 999 matrix
for which the entry in row and column is equal to
2
(

1000 ,

1000
)
.
Using this matrix, verify numerically that the function 2(,) is nonneg-
ative and satisfies ∫ 1
0
∫ 1
0
2(,)dd = 1.
What does this tell you about the function 2(,)?
The remaining questions will require you to work with the 999 × 999 matrix
specific to your student number. Call this matrix , and call its entries , where
is the row number and is the column number.
3
6. (4 points.) For ≥ 0 define the real-valued function () by
() = 1106
999∑︁
=1
999∑︁
=1
[

(

1000 ,

1000
)
−
]2
.
Use a “for loop” in Matlab to calculate (without displaying your answer)
the 201 × 1 vector whose -th entry is equal to

(
1 + − 1100
)
.
Using this vector and the plot command inMatlab, display a 2D graph
of the function () for between 1 and 3.
7. (1 point.) Which value of produces the minimum value of ()? (Confine
attention to the 201 values of at which you calculated () in the previous
question.) Let ∗ denote this minimizing choice of .
8. (1 point.) Calculate (without displaying your answer) the 999× 999 matrix
for which the entry in row and column is equal to
∗
(

1000 ,

1000
)
.
Using this matrix, verify numerically that the function ∗ (,) is nonneg-
ative and satisfies ∫ 1
0
∫ 1
0
∗ (,)dd = 1.
What does this tell you about the function ∗ (,)?
9. (2 points.) Let and be a continuous pair of random variables whose
joint PDF is ∗ (,). Display 2D graphs of the marginal PDFs for and
using the plot command inMatlab.
10. (2 points.) Calculate the covariance between the random variables and
introduced in the previous question.
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