Question

Some Extra Definitions Recall that, for a nonrandom real number c, and a random variable X, we have Var (cX) = e Var (X). In this problem we'll generalize this property to linear combinations! Let be a vector of real nonrandom numbers, and let be a vector of random variables (sometimes called a random vector). Last, define the covariance matrix to be the matrix with all the covariances ar- ranged into a matrix. When we talk about taking the taking the variance of a random vector, this is what we're talking about: [Cov(X1, X1) Cov(X1, X2) Cov(X1, X3)] Var (X) = = Cov(X2, X1) Cov(X2, X2) Cov(X2, X3) - Cov(X3, X1) Cov(X2, X2) Cov(X2, X3)]

Answer 1

Ans: 10Gbow that E(T X 2 = WTEX) we bave ECE OF C consider be a vertee of Redt numbee of constentofue det w=wiy 2 13] x = x17 *2 [*3] and vector of Random variable Of Random vertel Now considee FC WT x) = E[W, W z W3] 1x17 [x3] Wix,+ W 2X2 + ( 3X3 = E( X) + E(W2 X 2) + (W3X3) Muhafe gi uz spac Reul number of constent = W, E(X) +W2 E(X2)+W3E06) ECOX) = a (x) qis cons dent E (TX) - Wi E(Xi) +WzE(X2)+2E (X3) ....... I Lonside E WT Elx] = [wo Wz wz] Exq Now = {wi w2 W3] [E(x)] EX2) E(3) - W1 E(X) + 2 E(X2)+W 3 E(3) ..... TI from eibcive eah I and I E[WTx] = WTE(X) W' teangpose of werke w Hence prove

" To show that u w x ) = WT UU FLX ) W we have a is constant veaxl=a?ua Eck) by dep of variance. NOEL WTXI = E{(WT XD (WTX 1'] - [E(WTX)] 2 = E(wo Wz Wy] (*17 (wil [X1 X2 X3] [E(WTX)] - EWI(X, 7 {x1 X2 X3] (WT)] 1x2 - Į E(WTX) ? = EWT (WT)' XX'] - Te (WTx]2 = WTCWT)' E[xx'] - WTCWT) (E (X)]² - from liest. E(WTX)=WTECKI & Wil'=w VOEWT X) = WT W ELAX') - whole [x]] - wł w [ ECXX')-(E(x)]27 I = wiw varlxl. by defofuaeck) = WT UNE (X) W LUCE)ECXX') –(E ()] UAEWTXE WT UUE(X) W - Deove. and [wt] =twe=w Hence 12