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

Exercise 1.5.4 Let Y ∼ N(Jμ,σ 2I) and let O =

n−1/2J,O1 be an orthogonal matrix.

(a) Find the distribution of OY.

(b) Show that ¯ y· = (1/n)JY and that s2 =YO1O

1Y/(n−1).

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

Hint: Show that YY =YOOY =Y(1/n)JJY +YO1O

1Y.

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