MAST30025 Linear Statistical Models Assignment 1
Submission deadline: Friday March 24, 5pm
This assignment consists of 3 pages (including this page) with 5 questions and 33 total marks
Instructions to Students
Writing
This assignment is worth 6% of your total mark.
You may choose to either typeset your assignment in LATEX, or handwrite and scan it to
produce an electronic version.
You may use R for this assignment, but for matrix calculations only (you may not use the
lm function). If you do, include your R commands and output.
Write your answers on A4 paper. Page 1 should only have your student number, the
subject code and the subject name. Write on one side of each sheet only. Each question
should be on a new page. The question number must be written at the top of each page.
Scanning and Submitting
Put the pages in question order and all the same way up. Use a scanning app to scan all
pages to PDF. Scan directly from above. Crop pages to A4.
Submit your scanned assignment as a single PDF file and carefully review the submission
in Gradescope. Scan again and resubmit if necessary.
University of Melbourne 2023 Page 1 of 3 pages Can be placed in Baillieu Library
MAST30025 Linear Statistical Models Assignment 1 Semester 1, 2023
Question 1 (4 marks)
Prove that if a symmetric matrix A has eigenvalues which are all either 0 or 1, it is idempotent.
Question 2 (6 marks)
We wish to prove (without using Theorem 2.5) that if A, B, and A + B are n× n idempotent
matrices, then AB = BA = 0.
(a) Show that AB + BA = 0.
(b) By Theorem 2.2, there exists a matrix P which diagonalises A:
P TAP = D =
[
Ir 0
0 0
]
,
where r = r(A).
Write
P TBP = Λ =
[
Λ11 Λ12
Λ21 Λ22
]
,
where the block matrices are of the same dimensions as above. Show that Λ11 = 0, Λ12 = 0
and Λ21 = 0.
(c) Show that AB = BA = 0.
Question 3 (4 marks)
Show directly that for any random vector y and compatible matrix A, we have var Ay =
A(var y)AT .
Question 4 (7 marks)
Let y = (y1, y2, y3)T be a 3-dimensional multivariate normal random vector with mean and
variance .
(a) Find the eigenvalues and eigenvectors of V .
(b) Find the distribution of z1 = 3y1 + 2y2 + y3.
(c) Find the distribution of z2 = y21 +.
Page 2 of 3 pages
MAST30025 Linear Statistical Models Assignment 1 Semester 1, 2023
Question 5 (12 marks)
A secondary school teacher wants to know if the marks of students in Specialist Mathematics
can be predicted from their marks in General Mathematics. A linear model is assumed, and
the following data is obtained from nine students:
ID General Mathematics Specialist Mathematics
(a) Write down the linear model as a matrix equation, writing out the matrices in full.
(b) Calculate the least squares estimate of the parameters.
(c) Calculate the sample variance s2.
(d) Calculate the standardised residuals for all students.
(e) Calculate the Cook’s distances for all students.
(f) Predict (using a point estimate) the mark of Specialist Mathematics for a student whose
mark for General Mathematics is 90.
End of Assignment — Total Available Marks = 33