Page 364 - Applied Probability

Appendix B: The Normal Distribution
354
t
where Ω = LL is the variance matrix of Y .
To address the issue of conditional densities, consider the compatibly
partitioned vectors Y =(Y ,Y ), X =(X ,X ), µ =(µ ,µ ) and ma-
1
trices
t 1 t 2 t t t 2 t t t 1 t 2
L 11 0 Ω 11 Ω 12
L = Ω= .
L 21 L 22 Ω 21 Ω 22
Now suppose that X is standard normal, that Y = LX + µ, and that L 11
has full rank. For Y 1 = y 1 fixed, the equation y 1 = L 11 X 1 + µ 1 shows that
X 1 is fixed at the value x 1 = L −1 (y 1 − µ 1 ). Because no restrictions apply
11
to X 2 , we have
Y 2 = L 22 X 2 + L 21 L −1 (y 1 − µ 1 )+ µ 2 .
11
−1
Thus, Y 2 given Y 1 is normal with mean L 21 L 11 (y 1 − µ 1 )+ µ 2 and variance
t
t
L 22 L . To express these in terms of the blocks of Ω = LL , observe that
22
Ω 11 = L 11 L t 11
t
Ω 21 = L 21 L 11
t
Ω 22 = L 21 L t 21 + L 22 L .
22
The first two of these equations imply that L 21 L −1 =Ω 21 Ω −1 . The last
11 11
equation then gives
L 22 L t =Ω 22 − L 21 L t
22 21
t
=Ω 22 − Ω 21 (L ) −1 L −1
11 11 Ω 12
=Ω 22 − Ω 21 Ω −1 Ω 12 .
11
None of these calculations requires that Y 2 be of full rank. In summary, the
conditional distribution of Y 2 given Y 1 is normal with mean and variance
E(Y 2 | Y 1 )=Ω 21 Ω −1 (Y 1 − µ 1 )+ µ 2
11
Var(Y 2 | Y 1 )=Ω 22 − Ω 21 Ω −1 Ω 12 .
11
B.3 References

[1] Ciarlet PG (1989) Introduction to Numerical Linear Algebra and Op-
timization. Cambridge University Press, Cambridge
[2] Rao CR (1973) Linear Statistical Inference and its Applications, 2nd
ed. Wiley, New York