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

Exercise 1.5.6 Let Y = (y1,y2, y3) have a N(μ,σ 2I) distribution. Consider the quadratic forms defined by the matrices M1, M2, and M3 given below.

(a) Find the distribution of each YMiY.

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

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

M1 =

1 3

J3 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 ⎤
⎦.