PDF of Lecture Notes - School of Mathematical Sciences

1. DISTRIBUTION THEORY
Proof.
We will exhibit f(x 1 , x 2 ) in the form
f(x 1 , x 2 ) = h(x 1 , x 2 ) g(x 2 ),
where h(x 1 , x 2 ) is a PDF with respect to x 1 for each (given) x 2 .
It follows then that f X1 |X 2
(x 1 |x 2 ) = h(x 1 , x 2 ) and g(x 2 ) = f X2 (x 2 ).
Now observe that:
1
f(x 1 , x 2 ) =
(2π) (r 1+r 2 )/2
∣ Σ ∣ ×
11 Σ 12∣∣∣
1/2
Σ 12 Σ 22
and
exp
[
− 1 2
( (x1 ) T ( ) ) [ ] −1 ( ) ]
T Σ
− µ 1 , x2 − µ 11 Σ 12 x1 − µ 1
2
Σ 21 Σ 22 x 2 − µ 2
1.
(2π) (r 1+r 2 )/2
∣ Σ ∣
11 Σ 12∣∣∣
1/2
Σ 21 Σ 22
= (2π) r 1/2 (2π) r 2/2 ∣ ∣Σ 22
∣ ∣
1/2 ∣ ∣Σ 11 − Σ 12 Σ −1
22 Σ 21
∣ ∣
1/2
= (2π) r 1/2 ∣ ∣Σ 11 − Σ 12 Σ −1
22 Σ 21
∣ ∣
1/2
(2π)
r 2 /2 ∣ ∣Σ 22
∣ ∣
1/2
2.
(
(x1 − µ 1 ) T , (x 2 − µ 2 ) ) [ ] −1 ( )
T Σ 11 Σ 12 x1 − µ 1
= ( (x
Σ 21 Σ 22 x 2 − µ 1 − µ 1 ) T , (x 2 − µ 2 ) ) T
2
⎡
×
⎢
⎣
(
Σ11 − Σ 12 Σ −1
22 Σ 21
) −1
− ( Σ 11 − Σ 12 Σ −1
22 Σ 11
) −1
×Σ 12 Σ −1
22
−Σ −1
22 Σ 21 Σ −1
22 + Σ −1
22 Σ 21
× ( )
Σ 11 − Σ 12 Σ −1 −1
22 Σ 21 × ( )
Σ 11 − Σ 12 Σ −1 −1
22 Σ 21
×Σ 12 Σ −1
22
⎤
⎥
⎦
( )
x1 − µ 1
x 2 − µ 2
= (x 1 − µ 1 ) T V −1 (x 1 − µ 1 ) − (x 1 − µ 1 ) T V −1 Σ 12 Σ −1
22 (x 2 − µ 2 )
−(x 2 − µ 2 ) T Σ −1
22 Σ 21 V −1 (x 1 − µ 1 ) + (x 2 − µ 2 ) T Σ −1
22 Σ 21 V −1 Σ 12 Σ −1
22 (x 2 − µ 2 )
61
+(x 2 − µ 2 ) T Σ −1
22 (x 2 − µ 2 ).