Student
number
Semester 1 Assessment, 2022
School of Mathematics and Statistics
MAST30025 Linear Statistical Models Assignment 1
Submission deadline: Friday March 25, 5pm
This assignment consists of 3 pages (including this page) with 6 questions and 36 total marks
Instructions to Students
Writing
ˆ This assignment is worth 6% of your total mark.
ˆ You may choose to either typeset your assignment in LATEX, or handwrite and scan it to
produce an electronic version.
ˆ 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.
ˆ 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
ˆ 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.
ˆ 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 2022 Page 1 of 3 pages Can be placed in Baillieu Library
MAST30025 Linear Statistical Models Assignment 1 Semester 1, 2022
Question 1 (4 marks)
Let A be an idempotent n × n matrix. Find all scalars k such that the matrix I − kA is
idempotent.
Question 2 (5 marks)
Prove (without using Theorem 2.5) that if A and B are symmetric matrices, A+B is idempotent
and AB = BA = 0, then both A and B are idempotent. (Hint : Use Theorem 2.4. Then derive
two relations between the diagonalisations of A and B.)
Question 3 (10 marks)
Let y be a 3-dimensional multivariate normal random vector with mean and variance
µ =
 1−1
0
 , V =
 2 0 10 1 0
1 0 2
 .
Let
A = 13
 2 0 −10 3 0
−1 0 2
 .
(a) Describe the distribution of Ay.
(b) Find E[yTAy].
(c) Describe the distribution of yTAy.
(d) Find all matrices B such that yTBy is independent of yTAy.
Page 2 of 3 pages
MAST30025 Linear Statistical Models Assignment 1 Semester 1, 2022
Question 4 (4 marks)
Let AB = Im, where A is an m× n matrix and B is an n×m matrix. If y ∼MVN(µ, In) and
BA is symmetric, find the distribution of yTBAy.
Question 5 (8 marks)
A manager wants to know if (and how) the sales figures of each store is dependent on the
advertising cost and the size of the store. A linear model is assumed, and the following data is
obtained from six stores:
Sales ($k) Advertising cost ($k) Size (m2)
227 2 200
354 4 250
373 5 200
512 6 400
537 8 150
328 4 220
(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 residual sum of squares SSRes and sample variance s
2.
(d) Predict (using a point estimate) the average sales figure of a store with $3k advertising
cost and 350m2 size.
Question 6 (5 marks)
Let A be a symmetric and idempotent matrix with entries aij . Prove that 0 ≤ aii ≤ 1. Use this
to derive limits on the leverage of a point in the full rank model.
End of Assignment — Total Available Marks = 36
Page 3 of 3 pages