Page 254 - Elements of Distribution Theory

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052184472Xc08 CUNY148/Severini May 24, 2005 17:54

240 Normal Distribution Theory

absolutely continuous with density

d 1 1 T −1 d
(2π) − 2 | | − 2 exp − (x − µ) (x − µ) , x ∈ R .
2
Proof. Let Z be a d-dimensional vector of independent, identically distributed standard
normal random variables. Then Z has density

d

1 1 2 d 1 T d
exp − z = (2π) − 2 exp − z z , z ∈ R .
1 2 i 2
i=1 (2π) 2
1
By Theorem 8.1 part (vi), the density of X is given by the density of µ + 2 Z. Let
1
W = µ + 2 Z; this is a one-to-one transformation since | | > 0. Using the change-of-
variable formula, the density of W is given by

d 1 T −1
(2π) − 2 exp − (w − µ) (w − µ) ∂z .
2 ∂w
The result now follows from the fact that

1
∂z − 2 .
=| |

∂w
Example 8.5 (Bivariate normal distribution). Suppose X is a two-dimensional random
vector with a bivariate normal distribution, as discussed in Example 8.1. The parameters
of the distribution are the mean vector, (µ 1 ,µ 2 ), and the covariance matrix, which may be
written
σ 1 ρσ 1 σ 2
2
= 2 .
ρσ 1 σ 2 σ
2
The density of the bivariate normal distribution may be written
2
1 1 x 1 − µ 1
√ exp −
2
2πσ 1 σ 2 (1 − ρ ) 2(1 − ρ ) σ 1
2

2]
x 1 − µ 1 x 2 − µ 2 x 2 − µ 2
− 2ρ + ,
σ 1 σ 2 σ 2
2
for (x 1 , x 2 ) ∈ R .

8.3 Conditional Distributions

An important property of the multivariate normal distribution is that the conditional distri-
bution of one subvector of X given another subvector of X is also a multivariate normal
distribution.

Theorem 8.3. Let X be a d-dimensional random vector with a multivariate normal distri-
bution with mean µ and covariance matrix .