Question

= = 3, Cov(X1, X2) = 2, Cov(X2, X3) = -2, Let Var(X1) = Var(X3) = 2, Var(X2) Cov(X1, X3) = -1. i) Suppose Y1 = X1 - X2. Find Var(Y1). ii) Suppose Y2 = X1 – 2X2 – X3. Find Var(Y2) and Cov(Yı, Y2). Assuming that (X1, X2, X3) are multivariate normal, with mean 0 and covariances as specified above, find the joint density function fxı,Y,(y1, y2). iii) Suppose Y3 = X1 + X2 + X3. Compute the covariance matrix of (Y1, Y2, Y3)from that of (X1, X2, X3)7.

Answer 1

2, Var(X,) = 3 CoV ( XL, X3 ) = 2, cou(X2, X3)= -2 Let Vaulx, ) - Var(X3) COV (X, xz) -1 Suppose y = x - X₂ Find Varly ) RESULT: Var(ax+by) a Var(x) +b Var(y) + 2ab CON(X,Y) Here are given y = x - X2 1 Var ( Y, ) = Var( X, - X) - Vase (X, 1 + Var(x,) - 2 cov (X2, X, ) 2 + 3-(2x2) 5 - 4 Var (Y, ) = 1 Ans (1) Suppose Y = X, - 2X₂ - Xz. Find Var (ye) and cov (Y, Y2). Assuming that ( XL, Xg, X3 ) are multivariate normal with mean 1 0 and Covariances specified above find the joint density function (9,4) Var (Y) Var (X - 2X₂ - X₂) Vove (X.+-2.3Var (%3] +(-11° var (X3) + Q. 16-29 cov( XL2 Xe ) + 2.(-2).(-1) Lov (X2, X3) + 2 (-1) 1 Cov (X1, X3 ) Varl X, ) + 4 Var(X) + Var(X) - 4 COV(X,X2) + 4 coV( X2, X3 ) de - 2 Cov(X₂, X₂) 2 + 1483 )+ 9 - (4x2)+(4*(-2)) - (2x (-1)) 2+12+2-8-8 + 2 Var (Y) = Vare (X, -2X - Xg) = 2 And

CoV (Y, Ye ) = cov ( x - xe, X; -2X2 – xg) Cov( XA, Xu) - 2 Cove X, X, ) - CON(X2, X3) - Cov( X1, X2) + 2 cov ( Xy, X,) + Cov( X2, X3 ) Var(x,) - CoV ( X., X.,) - Cov (X., X3 ) - Cov( X, X, ) + & var (X2) + Cov (X2, X3] 2 - (2x2) -(-1) - 2 + (2X3) + (-2) 2-4 & L - 2 +6 - 2 = 1 COVEY: Ya ) 1 1 Ans (3 RESULT If X = (X, X, X3 ) a Nz ( , Egys) х Suppose 18 man bertationen then, and X ( where X (9] where helt lol 0 The Var (X2) COV(X, X2) COV{x1, x₂) Cov (X1, X3 ) E- COV(XL, X2) Var (X2) COV(Xq, X₂) Cov(x,, ) Veri (x3) COVCX,X₂) CoV (X2, X) & Var (X₂) 'Varl X 1 COV(X1, X2) (1) where Coul XL,X) VarlX2)

HE then, X" ~ N ( l , " Now Now, consider, Now, any li a, x; +9Xq +9₂ X3 y in is da random Variable the b, x, + b₂ X₂ + b₂ X₂ form of Random Variable of X1, X₂, X3 then y" w Neu a ,x, + 9 xq tazz vrector of where is mean b, x, + ba xz + b₂ Xz. (1) the Variance & Covariance (a,x,+ Qq X₂ + 9zX; matrix of Du has to Ib, x, + box + bgXg Now, Here X - Xq X - 2X2 X3 2 then E(Y) = E(X, - X, ) = E(X) - E(X2) - 0-0 =0 ELY) = E(X, -2*, - *3) - ECX,) - 2E(89) - E (X3) = 0-0-0 Var (Y) COVEY, Ya) 16 - 1.7. & verty COV(YY) Var (Y) (02 [ ] using equation B , & 1 . (% ,Y) w Ng (lith Ewl where er & { 18 a given to above. 1 le (Y) follow bivariate Normal drstrubution

Note Subbose u ( 4 ) { Cov(x4) Love COV(X,Y) 9 that, Corr (x,y)= COV(X,Y) ✓ Var (x). Voorlys COV(X,Y) Var (x) Varily COV(x, y) Cov(x, y) = goog Var (x) Varly, then, joint p.dif of x and y is given by 2(1-p?) Cecy 32-29 18:43 13-Ha) + (9-Mix)] f (x,y) X,Y 2o SI-ga Our Cases, 9 li and B Cov( Y.Y) - 1 COVEY, Y₂ ) Var (Y) Varty) den JE M = M = 0 Joint density function of Y and V is given by I (Ų, % ) Yg sta-4) (4)2-24(40)*:)=40303] e 21.4.52 LI- y 11 ette ( 238 + 20 S-1 Ja (99- 8,8 + -24,2 -~9474 260 f 19,42 Ang 271

Con(XX.) + cow (X,X, ) + CoV (X4,Xg) - Cov(X.Xg) - COV(X3, X)-CoV (X,X3) Subpose Yz = X, + x2 + xy . Combule the Co Voyance matrix of ů Ya Yz)from that of (X, X, X ₂) Covariance mateur of (Y, Yo Y₂ ) is [ Var (",) Co V (Ye, & cov ( 47.93) 2, & CoV (YY) Var (Y) cov (Ya, Y) cov (?, Y;) CON (1,1) Var (Y) part from co of (ii) & get. Var (Y) = 1 Var (Y) = 2 Covly, y) = 1 Vor (Y₃) = Var (X, + X2 + X₂) Var (X2) + Vorlxg) + Var(83) + 2 cou( X., Xg) +2 Covl X1, X3) + 2 cou (X2, X3) 2 + 3+2+(x2)+(2*(1)) +(2X(-2)) - 2+3+2+4-2-4 7 +4-2-4 = 5 Var (Y₃) = 5 CoV (Y, Yz) = cou(x, -X, X; +Xq + X₂) Vor (*] + covex, Xy ] + cov (X1, X3) - Cov (Xu, X, ) - Ver ( X ) - COV(X, X3) 2+(-1)-3-(-2) = 2-1-3+2 = 0 Cov(Y, Y) O

Now cov (Y,Y) = CoV (X-2X -X3, X, + xy + x3) CoV (X, XI ) + Cov(X, X ) +cov (Xy, X3) – 3 Cov(X1, X3 ) – 9 con Xe, X2) - 2 Coul X X ) CoV (XX) - CoN(X2, X) - CONCX3, X₂) Var (X:) + Cov( , ) + Cov(x, xg) - 9 cov( X2X2) - 2 Var (X2) 2 cov ( X2 X3 ) - Cov( x,.xg) - Cov(X2, X3) - You(x3) 2+2 + (-1) -(2x2) - (243) (2X(-2)) -(-1)-(-2)-2 = 2+2-1- 4-6 + 4 +1 +2-2 -2 COVEY, Y₂ ) = T Covariance matris of Therefore the (Y. Y, Y₂) is given by (var (y) COV (Y, Y) Lov (Yr, Y) 1 2 1) Ey COVEY, Y ) Var (Y) cou (Y2 %) -2 Ans COV(Y,143) CoV (YY) Var (₂)