STAT2004J – Linear Modelling Tutorial 1
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STAT2004J – Linear Modelling
Tutorial 1
Question 1
Calculate the following expressions
(a) 
(b) 
(c) For a1 , · · · , an 2 R and b1 , · · · , bn 2 R calculate 
(d) For a1 , · · · , an 2 R calculate 
Question 2
-be two- arbitrary sequences-of real numbers. De-note the averages of 
by X and Y , respectively, i.e., X := 
：
Xi and Y := 
：
Yi. Also deine SXY :=：
(Xi — X(-))(Yi — Y(-)) and SXX :=：
(Xi — X(-))2 .
(a) Show that SXY = ：
XiYi — nX(-)Y(-)
(b) Show that SXY = ：
Xi(Yi — Y(-)) =：
Yi(Xi — X(-))
(c) Show that SXX = ：
Xi(2) — nX(-)2
(d) The data below contain the reduction in blood pressure (Y) caused by an Antihypertensive drug at diferent doses (X). Calculate SXY , SYY and SXX for these data:
Question 3
The covariance of any two random variables X and Y is deined as
Cov(X, Y) = E((X - E(X))(Y - E(Y))).
Let a,b, c be ixed (deterministic) constants.
(a) Show that Cov(X, Y) = E(XY) - E(X)E(Y).
(b) Show that Cov(X, X) = Var(X) for any random variable X .
(c) Show that Cov(X, Y) = Cov(Y, X) for any random variables X, Y.
(d) Show that Cov(X + Y, Z) = Cov(X, Z) + Cov(Y, Z) for any random variables X, Y, Z.
(e) Show that Cov(a, X) = 0 for any deterministic constant a and any random variable X .
(f) Show that Cov(aX, bY) = abCov(X, Y) for any deterministic constants a, b and any ran- dom variables X, Y.
(g) Show that Cov(X, Y + c) = Cov(X, Y) for any deterministic constant c and any random variables X, Y.
(h) For any two sequences of random variables , 
and any two sequences of 
, 
, calculate Cov (Σ
aiXi , Σ
biYi).
(i) For any sequence of random variables {Xi}ni=1 and any sequence of deterministic real 
, calculate Var (Σ
aiXi).
where x and y denote the averages of the x and y sequences. Find the empirical covariance for the data set in Question 1.
2023-12-28