Exercise 1.5.6 Let Y = (y1, y2, y3) have a N(, 2 I ) distribution. Consider

Exercise 1.5.6 Let Y = (y1, y2, y3)

 have a N(μ, σ2 I ) distribution. Consider the quadratic forms defined by the matrices M1, M2, and M3 given below.

(a) Find the distribution of each Y MiY .

(b) Show that the quadratic forms are pairwise independent.

(c) Show that the quadratic forms are mutually independent.

M1 = 1 3

J 3 3 , M2 = 1 14

⎡

⎣

9 −3 −6

−3 1 2

−6 2 4

⎤

⎦, M3 = 1 42

⎡

⎣

1 −5 4

−5 25 −20 4 −20 16

⎤

⎦.