Summary of Some Useful Facts About Multivariate (Normal) Distributions
1. If X has mean vector μ and covariance matrix Σ , then Y = B X + d has mean vector
k ×1
k ×k
k ×1
l ×1
l ×k k ×1
l ×1
B μ + d and covariance matrix B Σ 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 + 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 ×1
k ×k
k × p p× k
any dimension p and any matrix A giving a particular Σ end up producing the same kdimensional joint distribution.)
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
⎝ l ×k k ×1 l ×1 l ×k k ×k k ×l ⎠
⎝ k ×1 k ×k ⎠
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
⎝ l ×1 l ×l ⎠
⎝ k ×1 k ×k ⎠
⎛ ⎛ μ1 ⎞ ⎛ Σ11 0 ⎞ ⎞
k ×l
⎛X⎞
k ×1
k ×k
⎟⎟ .
W = ⎜ ⎟ is MVN k +l ⎜ ⎜ ⎟ , ⎜
⎜
( k + l )×1
⎜
⎟
⎜
⎟⎟
Σ
μ
0
Y
⎝ ⎠
22 ⎟ ⎟
⎜ ⎜ 2 ⎟ ⎜ l ×k
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
k
1
−
−
⎛ 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 =⎜
⎜
⎟
k
×
k
⎜⎜ μ2 ⎟⎟
k ×1
22
⎜ ( kΣ−l21)×l ( k −Σ
⎟
−
×
(
k
l
)
1
l )×( k −l ) ⎠
⎝
⎠
⎝
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
−1
covariance matrix = Σ11 − Σ12Σ 22
Σ 21
8. All correlations between two parts of a MVN vector equal to 0 implies that those parts of the
vector are independent.