PDF of Lecture Notes - School of Mathematical Sciences

1. DISTRIBUTION THEORY
Finally, recall that if λ 1 , . . . , λ r are eigenvalues of Σ, then det(Σ) =
r∏
i=1
λ r
and
det(Σ) = 0
=⇒ det(Σ) > 0 for Σ positive definite,
for Σ non-negative definite but not positive definite.
1.10 The multivariable normal distribution
Definition. 1.10.1
The random vector X = (X 1 , . . . , X r ) T is said to have the r-dimensional multivariate
normal distribution with parameters µ ∈ R r and Σ r×r positive definite, if it has PDF
We write X ∼ N r (µ, Σ).
Examples
f X (x) =
1
(2π) r/2 |Σ| 1/2 e− 1 2 (x−µ)T Σ −1 (x−µ)
1. r = 2 The bivariate normal distribution
( ) [ ]
µ1
σ
2
Let µ = , Σ = 1 ρσ 1 σ 2
µ 2 ρσ 1 σ 2 σ2
2
=⇒ |Σ| = σ 2 1σ 2 2(1 − ρ 2 ) and
⎛
⎞
σ
Σ −1 1
2 2 −ρσ 1 σ 2
=
⎝
⎠
σ1σ 2 2(1 2 − ρ 2 )
−ρσ 1 σ 2 σ1
2
=⇒ f(x 1 , x 2 ) =
{[
1
√
2πσ 1 σ exp −1
2 1 − ρ
2 2(1 − ρ 2 )
[ (x1 ) 2 ( ) 2
− µ 1 x2 − µ 2
+
σ 1 σ 2
( ) ( )]]}
x1 − µ 1 x2 − µ 2
−2ρ
.
σ 1 σ 2
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