HO 3. Let X ~ N 2 and (a) Determine the distribution of Y - X. (b) Determine the distribution of Y2|Y1 = y1. 4. Consider X ~ N3(u, E) where X2|X1 = x1 ~ N(-21, 1) E(X1|X2 = 12) = -(12+1)/2 X3|X1 = $1, X2 = 12 ~ N(3x1 + 2x2 + 1, 1) . 2 (a) Compute / and E. (b) Determine the distribution of X3|X1 + X2 = x. 5. Given that X1 ~ Ni(0, 1) X2 X1 = $1 ~ N(1 + 1, 1) X3|X1 = $1, X2 = 12 ~ N(1 - 12, 1)5 TO (a) Compute / and C. (b) Determine the distribution of X3|X1 + X2 =x. 5. Given that X1 ~ N(0, 1) X2 X1 = x1 ~ N(1 + 21, 1) X3 X1 - 11, X2 - 12 ~ N(1 - 12, 1) (a) Find the distribution of X - (X1, X2, X3) . (b) Find the distribution of X2|X1 - 21, X3 - 13. (c) Find the distribution of 6. Let 6 -4 2 X ~ N3 2 10 -6 CO 2 -6 12 Let Yi = X] and Y2 - X1 + X2 and Y3|Yi = y1, Y2 - y2 ~ N(y, + y2, 5). Compute the distribution of (Y1, Y2, Y3) T. 7. Consider an i.i.d. sample of size n from the bivariate population with pdf f(1, X2) 020212 0182 0121 for $1, 22 2 0. (a) Find the MLE of 0 = (01, 02). (b) Find the Cramer-Rao lower bound and the asymptotic variance of 0. Down. 8. Consider an i.i.d. sample