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Question 1 For two jointly distributed random vectors Y 6 RP and ?

Edit: I see how this works now. If you could do #2 that would be great help!HW 3 2018-1.jpg

HW 3 2018-1.jpg

Question 1 For two jointly distributed random vectors Y 6 RP and ? E Rq, if for allj = 1, . . . ,p and
k = 1, . . . , q, Xj and Yk are uncorrelated. Show that CovUz, 17’) = 01,“. Question 2 Let XI, ..., in be independent Npflr, 2‘.) random vectors. For not all zero wl, . . . , w”, define
11.
y“, = Z’wiii.
1:1 Find 0 6 am and d 6 RP, such that (in — arc-HZ, — d) ~ xg. X1
Question 3 Let )1: 2 X2 be a random vector with the population covariance 2.
X3
(a) If
3 1 1
E : 1 3 1 ,
1 1 3 find (05, ,8), such that
COV(X1,X3 — (aXl + IBXQD = COV(X2,X3 — (0X1 + 5X2» = 0.
(b) For a general population covariance matrix 011 012 013
2: 021 £722 023 7 0'31 0'32 033 CE 5 denote 211 := [011 U12] and v = [013]. If 211 is invertible, derive the formula for the vector [ such
U21 U22 023 that
COV(X1, X3 * (0X1 + 5X2» : C0V(X2=X3 * (0X1 + 5X2» : 0-

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