Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

202 Some Special Distributions

In Section 3.5.1, we discussed the contours of the bivariate normal distribution.
We now extend that discussion to the general case, adding probabilities to the
contours. LetXhave aNn(μ,Σ) distribution. In then-dimensional case, the
contours of constant probability for the pdf ofX, (3.5.16), are the ellipsoids

(x−μ)′Σ−^1 (x−μ)=c^2 ,

forc>0. Define the random variableY =(X−μ)′Σ−^1 (X−μ). Then using
expression (3.5.12), we have

Y=Z′Σ^1 /^2 Σ−^1 Σ^1 /^2 Z=Z′Z=

∑n

i=1

Zi^2.

SinceZ 1 ,...,Znare iidN(0,1),Y hasχ^2 -distribution withndegrees of freedom.

Denote the cdf ofY byFχ (^2) n.Thenwehave
P[(X−μ)′Σ−^1 (X−μ)≤c^2 ]=P(Y≤c^2 )=Fχ (^2) n(c^2 ). (3.5.17)
These probabilities are often used to label the contour plots; see Exercise 3.5.5. For
reference, we summarize the above proof in the following theorem. Note that this
theorem is a generalization of the univariate result given in Theorem 3.4.1.
Theorem 3.5.1.SupposeXhas aNn(μ,Σ)distribution, whereΣis positive defi-
nite. Then the random variableY=(X−μ)′Σ−^1 (X−μ)has aχ^2 (n)distribution.
The following two theorems are very useful. The first says that a linear trans-
formation of a multivariate normal random vector has a multivariate normal distri-
bution.
Theorem 3.5.2.SupposeXhas aNn(μ,Σ)distribution. LetY=AX+b,where
Ais anm×nmatrix andb∈Rm.ThenYhas aNm(Aμ+b,AΣA′)distribution.
Proof:From (3.5.15), fort∈Rm,themgfofYis
MY(t)=E[exp{t′Y}]
= E[exp{t′(AX+b)}]
=exp{t′b}E[exp{(A′t)′X}]
=exp{t′b}exp{(A′t)′μ+(1/2)(A′t)′Σ(A′t)}
=exp{t′(Aμ+b)+(1/2)t′AΣA′t},
which is the mgf of anNm(Aμ+b,AΣA′) distribution.
A simple corollary to this theorem gives marginal distributions of a multivariate
normal random variable. LetX 1 be any subvector ofX, say of dimensionm<
n. Because we can always rearrange means and correlations, there is no loss in
generality in writingXas
X=
[
X 1
X 2
]
, (3.5.18)