Global Positioning System Reference
In-Depth Information
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The mean and the covariance matrix completely describe the multivariate normal
distribution. The notation
N
n
n
µ
1
,
n
Σ
n
n
x
1
∼
(4.218)
is used. The dimension of the distribution is
n
.
In the following, some theorems on multivariate normal distributions are given
without proofs. These theorems are useful in deriving the distribution of
v
T
Pv
and
some of the basic statistical tests in least-squares adjustments.
Theorem 1
If
x
is multivariate normal
x
∼
N(
µ
,
Σ
)
(4.219)
[13
and
z
=
m
D
n
x
(4.220)
Lin
—
-0.
——
Lon
*PgE
is
a linear function of the random variable, where
D
is a
m
×
n
matrix of rank
m
≤
n
,
then
N
m
D
D
T
z
∼
µ
,
D
Σ
(4.221)
is
a multivariate normal distribution of dimension
m
. The mean and variance of the
ra
ndom variable
z
follow from the laws for propagating the mean (4.33) and variance-
co
variances (4.34).
[13
Th
eorem 2
)
, the marginal distribution
of
any set of components of
x
is multivariate normal with means, variances, and
co
variances obtained by taking the proper component of
If
x
is multivariate normal
x
∼
N(
µ
,
Σ
µ
and
Σ
. For example, if
x
1
x
2
N
,
µ
1
µ
2
Σ
11
Σ
12
x
=
∼
(4.222)
Σ
21
Σ
22
th
en the marginal distribution of
x
2
is
N
µ
2
,
Σ
22
x
2
∼
(4.223)
Th
e same law holds, of course, if the set contains only one component, say
x
i
. The
m
arginal distribution of
x
i
is then
n
µ
i
,
i
2
x
i
∼
σ
(4.224)
Theorem 3
If
x
is multivariate normal, a necessary and sufficient condition that two
subsets of the random variables are stochastically independent is that the covariances
be zero. For example, if