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MATH2831 - Assignment 2 - 2023 T3

Question 2 [20 marks] - must be submitted as a group work on Moodle

You can work on this question as a group of up to four students.  Your answers can be typed (e.g. as a Latex

or Word document) or handwritten, then converted (printed) into a pdf file and uploaded on Moodle submission link. Only one solution per group must be submitted. Please make sure that you list names of all group

members and their zIDs on the cover page of your submitted document.

For full marks, answers need to be mathematically correct and written clearly using plain English.

In the box below, list names and zIDs of all group members.

Part A. [7 marks] Consider the general linear model

y = Xβ + ε,   (1)

where as usual y is an n × 1 vector of responses, X is an n × p design matrix, β is a p × 1 vector of parameters, and ε is a vector of uncorrelated errors with zero mean and variance σ2 .

Consider the following transformation of ε:

q = β + Lε,

where L is a real p × n matrix.

A1. Determine E(q), the mean vector of q, and Var(q), the covariance matrix of q.

Give your answers as expressions with only  β , σ 2 and L (you must justify your answers).

(Hint: Covariance matrix of a random vector Z is defined as Var(Z) = E[(Z − E(Z))(Z − E(Z))⊤ ]).

A2. Determine E(q⊤ q).