PDF of Lecture Notes - School of Mathematical Sciences

1. DISTRIBUTION THEORY
Now suppose Σ r×r is any symmetric positive definite matrix and recall that we can write
Σ = E ∧ E T where ∧ = diag (λ 1 , λ 2 , . . . , λ r ) and E r×r is such that EE T = E T E = I.
Since Σ is positive definite, we must have λ i > 0, i = 1, . . . , r and we can define the
symmetric square-root matrix by:
Σ 1/2 = E ∧ 1/2 E T where ∧ 1/2 = diag ( √ λ 1 , √ λ 2 , . . . , √ λ r ).
Note that Σ 1/2 is symmetric and satisfies:
( ) Σ
1/2 2
= Σ 1/2 Σ 1/2T = Σ.
Now recall that if Z 1 , Z 2 , . . . , Z r are i.i.d. N(0, 1) then Z = (Z 1 , . . . , Z r ) T ∼ N r (0, I).
Because of the i.i.d. N(0, 1) assumption, we know in this case that E(Z) = 0, Var(Z) =
I.
Now, let X = Σ 1/2 Z + µ:
1. From Theorem 1.9.9 we have
E(X) = Σ 1/2 E(Z) + µ = µ and Var(X) = Σ 1/2 Var(Z) ( Σ 1/2) T
.
2. From Theorem 1.10.1,
X ∼ N r (µ, Σ).
Since this construction is valid for any symmetric positive definite Σ and any
µ ∈ R r , we have proved,
Theorem. 1.10.2
If X ∼ N r (µ, Σ) then
Theorem. 1.10.3
E(X) = µ and Var(X) = Σ.
If X ∼ N r (µ, Σ) then Z = Σ −1/2 (X − µ) ∼ N r (0, I).
Proof.
Use Theorem 1.10.1.
Suppose
(
X1
X 2
)
∼ N r1 +r 2
((
µ1
µ 2
)
,
[ ])
Σ11 Σ 12
Σ 21 Σ 22
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