Problem 3.2 Let H ∈ R n×n be symmetric and idempotent, hence a projection matrix. Let x ∼ N(0, In). (a) Let σ > 0 be a positive number. Find the distribution of σx. (b) Let u = Hx and v = (I − H)x and find the joint distribution of (u, v). 1 (c) Someone claims that u and v are independent. Is that true? (d) Let µ ∈ Im(H). Show that Hµ = µ. (e) Assume that 1 ∈ Im(H) and find the distribution of 1 T Hx. Here, 1 = (1, . . . , 1) ∈ R n is the vector of all ones.

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