Page 255 - Elements of Distribution Theory

P1: JZP
052184472Xc08 CUNY148/Severini May 24, 2005 17:54

8.3 Conditional Distributions 241

Write X = (X 1 , X 2 ) where X 1 is p-dimensional and X 2 is (d − p)-dimensional, µ =
p
(µ 1 ,µ 2 ) where µ 1 ∈ R and µ 2 ∈ R d−p , and

11 12
=
21 22
where 11 is p × p, 12 = T is p × (d − p), and 22 is (d − p) × (d − p).
21
Suppose that | 22 | > 0. Then the conditional distribution of X 1 given X 2 = x 2 is a
multivariate normal distribution with mean vector
−1
µ 1 + 12 (x 2 − µ 2 )
22
and covariance matrix
−1
11 − 12 21 .
22
Proof. Let
I p − 12 22
−1
Z = X;
0 I q
then Z has a multivariate normal distribution with covariance matrix
−1

11 − 12 21 0
22
0 22
where q = d − p. Write Z = (Z 1 , Z 2 ) where Z 1 has dimension p and Z 2 has dimension q.
Note that Z 2 = X 2 . Then, by part (v) of Theorem 8.1, Z 1 and X 2 are independent. It follows
that the conditional distribution of Z 1 given X 2 is the same as the marginal distribution of
−1
Z 1 , multivariate normal with mean µ 1 − 12 µ 2 and covariance matrix
22
−1
11 − 12 21 .
22
Since
−1
Z 1 = X 1 − 12 X 2 ,
22
−1
X 1 = Z 1 + 12 22 X 2 ,
and the conditional distribution of X 1 given X 2 = x 2 is multivariate normal with mean given
by
−1 −1 −1 −1
E(Z 1 |X 2 = x 2 ) + 12 22 x 2 = µ 1 − 12 µ 2 + 12 22 x 2 = µ 1 + 12 (x 2 − µ 2 )
22
22
and covariance matrix
−1
11 − 12 21 ,
22
proving the theorem.
Example 8.6 (Bivariate normal). Suppose that X has a bivariate normal distribution, as
discussed in Example 8.5, and consider the conditional distribution of X 1 given X 2 = x 2 .
2
2
Then 11 = σ , 12 = ρσ 1 σ 2 , and 22 = σ .It follows that this conditional distribution
1 2
is normal, with mean
σ 1
µ 1 + ρ (x 2 − µ 2 )
σ 2