204 Some Special Distributionsfollowing proof shows, we can combine the results of Theorems 3.5.2 and 3.5.3 to
obtain the following theorem.
Theorem 3.5.4.SupposeXhas aNn(μ,Σ)distribution, which is partitioned as
in expressions (3.5.18) and (3.5.19). Assume thatΣis positive definite. Then the
conditional distribution ofX 1 |X 2 is
Nm(μ 1 +Σ 12 Σ− 221 (X 2 −μ 2 ),Σ 11 −Σ 12 Σ− 221 Σ 21 ). (3.5.22)Proof: Consider first the joint distribution of the random vectorW = X 1 −
Σ 12 Σ− 221 X 2 andX 2. This distribution is obtained from the transformation
[
W
X 2
]
=[
Im −Σ 12 Σ− 221
OIp][
X 1
X 2]
.Because this is a linear transformation, it follows from Theorem 3.5.2 that the joint
distribution is multivariate normal, withE[W]=μ 1 −Σ 12 Σ− 221 μ 2 ,E[X 2 ]=μ 2 ,
and covariance matrix
[
Im −Σ 12 Σ− 221
OIp
][
Σ 11 Σ 12
Σ 21 Σ 22][
Im O′
−Σ− 221 Σ 21 Ip]
=
[
Σ 11 −Σ 12 Σ− 221 Σ 21 O′
OΣ 22]
.Hence, by Theorem 3.5.3 the random vectorsWandX 2 are independent. Thus
the conditional distribution ofW|X 2 is the same as the marginal distribution of
W;thatis,
W|X 2 isNm(μ 1 −Σ 12 Σ− 221 μ 2 ,Σ 11 −Σ 12 Σ− 221 Σ 21 ).Further, because of this independence,W+Σ 12 Σ− 221 X 2 givenX 2 is distributed as
Nm(μ 1 −Σ 12 Σ− 221 μ 2 +Σ 12 Σ− 221 X 2 ,Σ 11 −Σ 12 Σ− 221 Σ 21 ), (3.5.23)which is the desired result.
In the following remark, we return to the bivariate normal using the above
general notation.
Remark 3.5.1 (Continuation of the Bivariate Normal). Suppose (X, Y)hasa
N 2 (μ,Σ) distribution, where
μ=[
μ 1
μ 2]
andΣ=[
σ^21 σ 12
σ 12 σ^22]. (3.5.24)
Substitutingρσ 1 σ 2 forσ 12 inΣ, it is easy to see that the determinant ofΣis
σ 12 σ^22 (1−ρ^2 ). Recall thatρ^2 ≤1. For the remainder of this remark, assume that
ρ^2 <1. In this case,Σis invertible (it is also positive definite). Further, sinceΣis
a2×2 matrix, its inverse can easily be determined to be
Σ−^1 =1
σ^21 σ 22 (1−ρ^2 )[
σ^22 −ρσ 1 σ 2
−ρσ 1 σ 2 σ 12]. (3.5.25)