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378 Appendix A Random Variables and Probability Distributions
Definition A.3.1 X has a multivariate normal distribution with mean µ and nonsingular covariance
matrix XX, written as X ∼ N(µ,),if
fX(x) (2π) −n/2 (det ) −1/2
exp − 1
2 (x − µ)′ −1
(x − µ) .
If X ∼ N(µ,), we can define a standardized random vector Z by applying the
linear transformation
Z −1/2 (X − µ), (A.3.1)
where −1/2 is defined as in the remark of Section A.2. Then by (A.2.4) and (A.2.5),
Z has mean 0 and covariance matrix ZZ −1/2 −1/2 In, where In is the n × n
identity matrix. Using the change of variables formula for probability densities (see
Mood, Graybill, and Boes, 1974), we find that the probability density of Z is
fZ(z) (det ) 1/2
1/2
fX z + µ
(det ) 1/2 (2π) −n/2 (det ) −1/2 exp
− 1
2 (−1/2z) ′ −1 −1/2
z
(2π) −n/2
exp − 1
2 z′
z
(2π) −1/2
exp − 1
2 z2
1
··· (2π) −1/2
exp − 1
2 z2
n
,
showing, by (A.2.2), that Z1,...,Zn are independent N(0, 1) random variables. Thus
the standardized random vector Z defined by (A.3.1) has independent standard normal
random components. Conversely, given any n×1 mean vector µ, a nonsingular n×n
covariance matrix , and an n×1 vector of standard normal random variables, we can
construct a normally distributed random vector with mean µ and covariance matrix
by defining
X 1/2 Z + µ. (A.3.2)
(See Problem A.4.)
Remark 1. The multivariate normal distribution with mean µ and covariance matrix
can be defined, even when is singular, as the distribution of the vector X in (A.3.2).
The singular multivariate normal distribution does not have a joint density, since
the possible values of X−µ are constrained to lie in a subspace of R n with dimension
equal to rank().