PDF of Lecture Notes - School of Mathematical Sciences

1. DISTRIBUTION THEORY
f(x 1 , x 2 ) =
1
(2π) (r 1+r 2 )/2
∣ Σ ∣
11 Σ 12∣∣∣
1/2
Σ 21 Σ 22
=
⎧
⎡ ⎤
⎨
Σ 11 Σ 12
× exp
⎩ −1 2 ((x 1 − µ 1 ) T , (x 2 − µ 2 ) T ) ⎣ ⎦
Σ 22
1
(2π) r 1/2
|Σ 11 − Σ 12 Σ −1
22 Σ 21 | 1/2
Σ 21
{
× exp − 1 2 (x 1 − (µ 1 + Σ 12 Σ −1
22 (x 2 − µ 2 ))) T
⎞⎫
x 1 − µ 1 ⎬
⎝ ⎠
⎭
x 2 − µ 2
−1 ⎛
× ( ) }
Σ 11 − Σ 12 Σ −1 −1(x1
22 Σ 21 − (µ 1 + Σ 12 Σ −1
22 (x 2 − µ 2 )))
×
{
1
(2π) ∣ ∣ r 2/2 ∣Σ exp − 1 }
22 1/2
2 (x 2 − µ 2 ) T Σ −1
22 (x 2 − µ 2 )
= h(x 1 , x 2 )g(x 2 ), where for fixed x 2 , h(x 1 , x 2 ), is the
N r1
(
µ1 + Σ 12 Σ −1
22 (x 2 − µ 2 ), Σ 11 − Σ 12 Σ −1
22 Σ 21
)
PDF, and g(x 2 ) is the N r2 (µ 2 , Σ 22 ) PDF.
Hence, the result is proved.
Theorem. 1.10.5
Suppose that X ∼ N r (µ, Σ) and Y = AX + b, where A p×r with linearly independent
rows and b ∈ R p are fixed. Then Y ∼ N p (Aµ + b, AΣA T ).
[Note: p ≤ r.]
Proof. Use method of regular transformations.
Since the
(
rows of A are linearly independent, we can find B (r−p)×r such that the r × r
B
matrix is invertible.
A)
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