Question 4 Let X N N3(u, E), Where

H = 71 , E 2 Which of the following random variables are independent? Explain.

(a) X1 and X2 (b) X1 and X3 (c) X2 and X3 (d) X1 and X1 + 3X2 — 2X3 Question 5 Suppose the joint p.d.f. of X and Y is rm): [2799:1303 [a3)2+<y+2)2+(m~3)<y+2>]), Find Z : h(X,Y), such that Z N X3- Question 6 Let X" be a random vector with positive deﬁnite population covariance matrix 2.

(a) What is the population covariance matrix of i7 = ETiX? (b) Let 2 2 L1 Aj'vjv; be the spectral decomposition, which implies that v1,...,vp are unit and

pairwise orthogonal. Let Y} = 1);}? for allj = 1,...,p and i7 = [Y1,...,Y},]T. Find the population covariance matrix of 17.15 it diagonal? Explain. Question 7 Let u], . . . ,uq E R19 be unit and pairwise orthogonal vectors (q S p). Let Z G N(0, Ip). _. (a) Find the joint distribution of nil—Z, ..., “III—Z; (b) Find the distribution of iT( 31:1 wuﬂz.