Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

3.5. The Multivariate Normal Distribution 201

Note thatΣ^1 /^2 is symmetric and positive semi-definite. SupposeΣis positive
definite; that is, all of its eigenvalues are strictly positive. Based on this, it is then
easy to show that (

Σ^1 /^2

)− 1

=Γ′Λ−^1 /^2 Γ; (3.5.11)

see Exercise 3.5.13. We write the left side of this equation asΣ−^1 /^2. These matrices
enjoy many additional properties of the law of exponents for numbers; see, for
example, Arnold (1981). Here, though, all we need are the properties given above.
SupposeZhas aNn( 0 ,In) distribution. LetΣbe a positive semi-definite,
symmetric matrix and letμbe ann×1 vector of constants. Define the random
vectorXby
X=Σ^1 /^2 Z+μ. (3.5.12)
By (3.5.6) and Theorem 2.6.3, we immediately have

E[X]=μand Cov[X]=Σ^1 /^2 Σ^1 /^2 =Σ. (3.5.13)

Further, the mgf ofXis given by

MX(t)=E[exp{t′X}]=E

[

exp{t′Σ^1 /^2 Z+t′μ}

]

=exp{t′μ}E

[

exp

{(

Σ^1 /^2 t

)′

Z

}]

=exp{t′μ}exp

{

(1/2)

(

Σ^1 /^2 t

)′

Σ^1 /^2 t

}

=exp{t′μ}exp{(1/2)t′Σt}. (3.5.14)

This leads to the following definition:
Definition 3.5.1(Multivariate Normal).We say ann-dimensional random vector
Xhas amultivariate normal distributionif its mgf is

MX(t)=exp{t′μ+(1/2)t′Σt}, for allt∈Rn. (3.5.15)

whereΣis a symmetric, positive semi-definite matrix andμ∈Rn. We abbreviate
this by saying thatXhas aNn(μ,Σ)distribution.

Note that our definition is for positive semi-definite matricesΣ. UsuallyΣis
positive definite, in which case we can further obtain the density ofX.IfΣis
positive definite, then so isΣ^1 /^2 and, as discussed above, its inverse is given by
expression (3.5.11). Thus the transformation betweenXandZ, (3.5.12), is one-to-
one with the inverse transformation

Z=Σ−^1 /^2 (X−μ)

and the Jacobian|Σ−^1 /^2 |=|Σ|−^1 /^2. Hence, upon simplification, the pdf ofXis
given by

fX(x)=

1

(2π)n/^2 |Σ|^1 /^2

exp

{

−

1

2

(x−μ)′Σ−^1 (x−μ)

}

, forx∈Rn. (3.5.16)