Summary of Some Useful Facts About Multivariate Normal Distributions
1. The MVN distribution is most usefully defined as the distribution of X = A Z + µ 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 µ and covariance matrix Σ = A × A ′ . (This definition turns out to be unambiguous
k ×k
k ×1
k× p
p ×k
… any dimension p and any matrix A giving a particular Σ end up producing the same kdimensional joint distribution.)
2. (Not explicitly said in class … only the MVN version was stated) If X has mean
k ×1
vector µ and covariance matrix Σ , then Y = B X + d has mean vector B µ + d and
k ×k
k ×1
l ×1
l ×k k ×1
l ×1
l ×k k ×1
l ×1
covariance matrix B Σ B ′ .
l ×k k × k k × l
3. If X is MVN k  µ , Σ  then Y = B X + d is MVN l  B µ + d , B Σ B ′  .
k ×1
l ×1
l ×k k ×1 l ×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 µ and
k ×1
covariance matrix the appropriate sub-matrix of Σ ).
k ×k
5. If X is MVN k  µ1 , Σ11  and independent of Y which is MVN l  µ2 , Σ 22  , then the vector
k ×1
l ×1
 k ×1 k ×k 
 l ×1 l ×l 
  µ1   Σ11 0  
k ×l
X
k×
k ×k
 .
W =   is MVN k + l   1  , 
( k + l )×1




Y 
  µ2   l0×k Σ 22  
l ×l  
  l ×1  
6. For non-singular Σ the MVN k  µ , Σ  distribution has a (joint) pdf on k-dimensional space
k ×k
 k ×1 k ×k 
given by
1
k
−
−
 1

f X ( x ) = ( 2π ) 2 det Σ 2 exp  − ( x − µ )′ Σ −1 ( x − µ ) 
 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  µ , Σ  where
k ×1
 X 2 
 k ×1 k ×k 
 ( k −l )×1 
Σ12 
 Σ11
 µ1 
l ×l
l ×( k −l ) 
l ×1 


and Σ =
µ=
 Σ 21

k ×k
 µ2 
k ×1
22
 ( k −l )×l ( k −Σ

l )×( k −l ) 
 ( k −l )×1 

the conditional distribution of X 1 given that X 2 = x2 is MVN l with
−1
mean vector = µ1 + Σ12Σ 22
( x2 − µ 2 )
and
covariance matrix = Σ11 − Σ12 Σ−221Σ21
8. All correlations between two parts of a MVN vector equal to 0 implies that those parts of the
vector are independent.