Global Positioning System Reference
In-Depth Information
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These integrals are zero because of the definition of the mean. The converse, i.e., zero
correlation, implies stochastic independence is valid only for the multivariate normal
distribution.
Variance-Covariance Matrix
Equations (4.20) and (4.21) can be used to com-
pute the variances and covariances for all components in the random vector
x
. Arrang-
ing the result in the form of a matrix yields the variance-covariance matrix. Thus, for
the random vector
−
µ
x
=
˜
x
n
− µ
x
n
T
x
x
1
− µ
x
1
···
˜
(4.25)
the
(n
×
n)
variance-covariance matrix becomes
[10
x
1
σ
σ
x
1
,x
2
···
σ
x
1
,x
n
E
x
−
µ
x
T
···
σ
−
µ
x
x
x
2
,x
n
Lin
—
-4.
——
Nor
PgE
Σ
x
=
=
(4.26)
.
.
.
.
x
n
sym
σ
Th
e variance-covariance matrix is symmetric because of (4.21). The expectation
op
erator
E
is applied to each matrix element. The variance-covariance matrix is
si
mply called the covariance matrix for the sake of brevity. The correlations are
co
mputed according to Equation (4.22) and can be arranged in the same order. Thus,
th
e correlation matrix is
[10
1
ρ
x
1
,x
2
···
ρ
x
1
,x
n
···
ρ
x
2
,x
n
C
=
(4.27)
.
.
.
.
sym
1
The correlation matrix is symmetric, the diagonal elements equal 1, and the off-
diagonal elements are between
−
1 and
+
1.
Propagation
Usually we are more interested in a linear function of the random
variables than in the random variables themselves. Typical examples are the adjusted
coordinates used to compute distances and angles. From the definition of the mean
(4.11), it follows that for a constant
c
c
∞
−∞
E(c)
=
f(x)dx
=
c
(4.28)
and
