Question

Q2) All sub problems are related. Show all steps for full credit. Let U and V be independent and identically distributed (i.i.d.) Gaussian(0,2) (mean = 0, and standard deviation 2) random variables. The (2x1) random vector X is given as X = II a) Find the covariance matrix of the random vector X. (10 points) . Find the expected value b) A (2x1) derived random vector Y = 2 is given as Y = AX where A = [1 vector and covariance matrix of the random vector Y. (10 points) c) Are the transformed random variables Y and Y2 independent? Justify your answer for full credit. (Hint: Answer first whether or not Y, and Y, are jointly Gaussian.) (5 points)

Answer 1

solution : oriven that Let uand v be independent and identically distributed ci.i.d) standard craussia nco, a) (meanzo and deviation=2) random variables car criven u and V ore independent oraussian cos Let x = [ų ] Then mean of the vector x is given by Ex= [eruit = [:)

The covariance matrix of x is given by to {= vcx 2 vaglun Lorcu,vd varcu, v vcu) © 14 Oy Hence X has by variance oraus-san distribu- tion with mean vector o and covariance matorine E C62. Let Y = AX Then as x has ouassian distribution y also has oraussian distribution with mean ECY ? = AE CX) = [ o ] and covariance matrix VCY= AVCXDA bet = A EA?

= ri 1784 oni 4 JL i 1] + [:-][:][-] = + [:-] [:-] we have Y = AX = (-3 [] = utvy y * [] LY2 since x is caussian ry 7 = Y = AX is also a Gaussian distribution as linear combination of oraussian is also caussian (62 From aort (b), we have VCY= v ry, y

CONCY,,72) var (727) Кетсе, cov CY,,Y22=0 and [% impres each component Y ] is Gaussian and Yg is oaussian since, zero covariance implies independence for oraussian random variables y, and Ye are independent - X Thank you