Question: Exercise 1.5.4 Let Y N(J, 2 I ) and let O = n1/2 J, O1 be an orthonormal matrix. (a) Find the distribution

Exercise 1.5.4 Let Y ∼ N(Jμ, σ2 I ) and let O =

n−1/2 J, O1



be an orthonormal matrix.

(a) Find the distribution of OY .

(b) Show that ¯y· = (1/n)J Y and that s2 = Y O1O

1Y/(n − 1).

(c) Show that ¯y· and s2 are independent.

Hint: Show that Y Y = Y OOY = Y 

(1/n)J JY + Y O1O

1Y .

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