Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

198 Some Special Distributions

3.5 TheMultivariateNormalDistribution

In this section we present the multivariate normal distribution. In the first part

of the section, we introduce the bivariate normal distribution, leaving most of the

proofs to the later section, Section 3.5.2.

3.5.1 BivariateNormalDistribution..................

We say that (X, Y) follows abivariate normal distributionif its pdf is given by

f(x, y)=

1

2 πσ 1 σ 2

√

1 −ρ^2

e−q/^2 , −∞<x<∞, −∞<y<∞, (3.5.1)

where

q=

1

1 −ρ^2

[(

x−μ 1

σ 1

) 2

− 2 ρ

(

x−μ 1

σ 1

)(

y−μ 2

σ 2

)

+

(

y−μ 2

σ 2

) 2 ]

, (3.5.2)

and−∞<μi<∞,σi>0, fori=1,2, andρsatisfiesρ^2 <1. Clearly, this function
is positive everywhere inR^2. As we show in Section 3.5.2, it is a pdf with the mgf
given by:

M(X,Y)(t 1 ,t 2 )=exp

{

t 1 μ 1 +t 2 μ 2 +

1

2

(t^21 σ 12 +2t 1 t 2 ρσ 1 σ 2 +t^22 σ^22 )

}

. (3.5.3)

Thus, the mgf ofXis

MX(t 1 )=M(X,Y)(t 1 ,0) = exp

{

t 1 μ 1 +

1

2

t^21 σ 12

}

;

hence,Xhas aN(μ 1 ,σ 12 ) distribution. In the same way,Yhas aN(μ 2 ,σ 22 ) distri-
bution. Thusμ 1 andμ 2 are the respective means ofXandYandσ^21 andσ^22 are the
respective variances ofXandY. For the parameterρ, Exercise 3.5.3 shows that

E(XY)=

∂^2 M(X,Y)

∂t 1 ∂t 2

(0,0) =ρσ 1 σ 2 +μ 1 μ 2. (3.5.4)

Hence, cov(X, Y)=ρσ 1 σ 2 and thus, as the notation suggests,ρis the correlation
coefficient betweenXandY. We know by Theorem 2.5.2 that ifXandY are
independent thenρ= 0. Further, from expression (3.5.3), ifρ= 0 then the joint
mgfof(X, Y) factors into the product of the marginal mgfs and, hence,XandYare
independent random variables. Thus if (X, Y) has a bivariate normal distribution,
thenXandYare independent if and only if they are uncorrelated.
The bivariate normal pdf, (3.5.1), is mound shaped overR^2 and peaks at its
mean (μ 1 ,μ 2 ); see Exercise 3.5.4. For a givenc>0, the points of equal probability
(or density) are given by{(x, y):f(x, y)=c}. It follows with some algebra that
these sets are ellipses. In general for multivariate distributions, we call these sets
contoursof the pdfs. Hence, the contours of bivariate normal distributions are