Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

3.5. The Multivariate Normal Distribution 199

elliptical. IfXandY are independent then these contours are circular. The in-
terested reader can consult a book on multivariate statistics for discussions on the
geometry of the ellipses. For example, ifσ 1 =σ 2 andρ>0, the main axis of the
ellipse goes through the mean at a 45oangle; see Johnson and Wichern (2008) for
discussion.
Figure 3.5.1 displays a three-dimensional plot of the bivariate normal pdf with
(μ 1 ,μ 2 )=(0,0),σ 1 =σ 2 =1,andρ=0.5. For location, the peak is at (μ 1 ,μ 2 )=
(0,0). The elliptical contours are apparent. Locate the main axis. For a regionA
in the plane,P[(X, Y)∈A] is the volume under the surface overA. In general such
probabilities are calculated by numerical integration methods.

x

y

z

Figure 3.5.1:A sketch of the surface of a bivariate normal distribution with mean

(0,0),σ 1 =σ 2 =1,andρ=0.5.

In the next section, we extend the discussion to the general multivariate case;
however, Remark 3.5.1, below, returns to the bivariate case and can be read with
minor knowledge of vector and matrices.

3.5.2 ∗Multivariate Normal Distribution, General Case

In this section we generalize the bivariate normal distribution to then-dimensional

multivariate normal distribution. As with Section 3.4 on the normal distribution,

the derivation of the distribution is simplified by first discussing the standardized

variable case and then proceeding to the general case. Also, in this section, vector

and matrix notation are used.