PDF of Lecture Notes - School of Mathematical Sciences

1. DISTRIBUTION THEORY
If we take Z = BX, we have
( ( ( Z B 0
= X + ∼ N
Y)
A)
b)
by Theorem 1.10.1.
([ ] [ ])
Bµ + 0 BΣB
T
BΣA
,
T
Aµ + b AΣB T AΣA T
Hence, from Theorem 1.10.4, the marginal distribution for Y is
N p (Aµ + b, AΣA T ).
1.10.1 The multivariate normal MGF
The multivariate normal moment generating function for a random vector X ∼ N(µ, Σ)
is given by
M X (t) = e tT µ+ 1 2 tT Σt
Prove this result as an exercise!
The characteristic function of X is
E[exp(it T X)] = exp
(
it T µ − 1 )
2 tT Σt
The marginal distribution of X 1 (or X 2 ) is easy to derive using the multivariate normal
MGF.
Let
t =
(
t1
t 2
)
, µ =
(
µ1
µ 2
)
.
Then the marginal distribution of X 1 is obtained by setting t 2 = 0 in the expression
for the MGF of X.
Proof:
(
M X (t) = exp t T µ + 1 )
2 tT Σt
= exp
(t T 1 µ 1 + t T 2 µ 2 + 1 2 t 1 T Σ 11 t 1 + t T 1 Σ 12 t 2 + 1 )
2 t 2 T Σ 22 t 2 .
Now,
( )
t1
M X1 (t 1 ) = M X = exp
(t T
0
1 µ 1 + 1 )
2 t 1 T Σ 11 t 1 ,
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