Summary of Some Useful Facts About Multivariate (Normal) Distributions
1.
If X has mean vector  and covariance matrix  , then Y  B X  d has mean vector
k 1
k k
k 1
l 1
l k k 1
l 1
B   d and covariance matrix B  B  .
l k k 1
l 1
l k k k k l
2. The MVN distribution is most usefully defined as the distribution of X  A Z   for Z a
k 1
k  p p1
p1
k 1
vector of independent standard normal random variables. Such a random vector has mean
vector  and covariance matrix   A  A  . (This definition turns out to be unambiguous
k 1
k k
k p
p k
… any dimension p and any matrix A giving a particular  end up producing the same kdimensional joint distribution.)
3. If X is MVN k   ,   then Y  B X  d is MVN l  B   d , B  B   .
l 1
l k k 1 l 1
k 1
 k 1 k k 
 l k k 1 l 1 l k k k k l 
4. If X is MVN k , its individual marginal distributions are univariate normal. Further, any subvector of dimension l  k is MVN l (with mean vector the appropriate sub-vector of  and
k 1
covariance matrix the appropriate sub-matrix of  ).
k k
5. If X is MVN k  1 , 11  and independent of Y which is MVN l  2 , 22  , then the vector
l 1
k 1
 k 1 k k 
 l 1 l l 
  1   11 0  
k l
X
k 1
k k
 .
W    is MVN k l    , 

( k  l )1





0

Y
 
22  
  2   l k
l l  
  l 1  
6. For non-singular  the MVN k   ,   distribution has a (joint) pdf on k-dimensional space
k k
 k 1 k k 
given by
k
1


 1

f X ( x )   2  2 det  2 exp    x     1  x    
 2

7. The joint pdf given in 6 above can be studied and conditional distributions (given values for
 X1 
l 1 
part of the X vector) identified. For X  
MVN k   ,   where
k 1
 X 2 
 k 1 k k 
 ( k l )1 
12 
 11
 1 
l

l
l

( k l ) 
k

1

 and  
 

k

k
  2 
 22 
k 1
 21
(
k

l
)

1


 ( k l )l ( k l )( k l ) 
the conditional distribution of X1 given that X 2  x2 is MVN l with
1
mean vector  1  1222
 x2  2 
and
1
covariance matrix  11  1222
21
8. All correlations between two parts of a MVN vector equal to 0 implies that those parts of the
vector are independent.