Question: Please help with problem. Thank you! Problem 3.2 Let H E Rnx be symmetric and idempotent, hence a projection matrix. Let x ~ N(O, In).

Please help with problem. Thank you!

Problem 3.2 Let H E Rnx" be symmetric and idempotent, hence a projection matrix. Let x ~ N(O, In). (a) Let o > 0 be a positive number. Find the distribution of ox. (b) Let u = Hx and v = (I - H)x and find the joint distribution of (u, v).(c) Someone claims that u and v are independent. Is that true? (d) Let ME Im(H). Show that Hu = M. (e) Assume that 1 6 Im(H) and find the distribution of 1 Hx. Here, 1 = (1, . . ., 1) ER" is the vector of all ones

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