PDF of Lecture Notes - School of Mathematical Sciences

=⇒ Var(X) = Σ =
=⇒ Σ −1 =
=⇒ f X (x) =
=
=
⎡
⎤
σ 11 0 . . . 0
0 σ 22 . . . .
⎢
⎣ .
. .. . ⎥ .. . ⎦
0 . . . 0 σ rr
⎡
σ −1
⎤
11 0 . . . 0
0 σ −1
22 . . . .
⎢
⎣ .
. .. . ⎥ .. . ⎦
0 . . . 0 σrr
−1
1
(2π) r/2 |Σ| 1/2 e− 1 2 (x−µ)T Σ −1 (x−µ)
1. DISTRIBUTION THEORY
and |Σ| = σ 11 σ 22 . . . σ rr
1
1 P r (x i −µ i ) 2
(2π) r/2 (σ 11 σ 22 . . . σ rr ) 1/2 e− 2 i=1 σ ii
(
) (
1
√ √ e − 1 (x 1 −µ 1 ) 2
1
2 σ 11 √ √ e − 1 2
2π σ11 2π σ22
)
(x 2 −µ 2 ) 2
σ 22 . . .
(
)
1
. . . √ √ e − 1 (xr−µr) 2
2 σrr
2π σrr
= f 1 (x 1 )f 2 (x 2 ) . . . f r (x r )
=⇒ X 1 , . . . , X r are independent.
Note:
(x − µ) T Σ −1 (x − µ) = (x 1 − µ 1 x 2 − µ 2 . . . x r − µ r )
=
⎡
σ −1
⎤ ⎡ ⎤
11 0 . . . 0 x 1 − µ 1
0 σ22 −1
. . . .
x 2 − µ 2
× ⎢
⎥ ⎢ ⎥
⎣ .
.. .
.. . . ⎦ ⎣ . ⎦
0 . . . 0 σrr
−1 x r − µ r
r∑ (x i − µ i ) 2
.
σ ii
i=1
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