Appendix B: Multivariate Normal Distributions
A k-dimensional random vector 
follows a multivariate normal distribution with mean 
and positive-definite covariance matrix 
if its probability density function (pdf) is
8.47 
We use the notation 
to denote that 
follows such a distribution. This normal distribution plays an important role in multivariate statistical analysis and it has several nice properties. Here we consider only those properties that are relevant to our study. Interested readers are referred to Johnson and Wichern (1998) for details.
To gain insight into multivariate normal distributions, consider the bivariate case (i.e., k = 2). In this case, we have
Using the correlation coefficient ρ = σ12/(σ1σ2), where 
is the standard deviation of xi, we have 
and 
. The pdf of 
then becomes
where
Chapter 4 of Johnson and Wichern (1998) contains some plots of this pdf function.
Let 
be a nonzero k-dimensional vector. Partition the random vector as 
, where 
and 
with 1 ≤ p < k. Also partition 
and 
accordingly as
Some properties of 
are as follows:
1. 
. That is, any nonzero linear combination of 
is univariate normal. The inverse of this property also holds. Specifically, if 
is univariate normal for any nonzero vector 
, then 
is multivariate normal.
2. The marginal distribution of 
is normal. In fact, 
for i = 1 and 2, where k1 = p and k2 = k − p.
3. 
if and only if 
and 
are independent.
4. The random variable 
follows a chi-squared distribution with m degrees of freedom.
5. The conditional distribution of 
given 
= 
is also normally distributed as
The last property is useful in many scientific areas. For instance, it forms the basis for time series forecasting under the normality assumption and for recursive least-squares estimation.