Global Positioning System Reference
In-Depth Information
∞
...
∞
−∞
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µ
x
i
=
E(
x
i
)
˜
=
x
i
f
(
x
1
,x
2
,...,x
n
)
dx
1
dx
2
···
dx
n
(4.18)
−∞
In vector notation the expected values of all parameters are
=
E(
x
n
)
T
E(
x
)
x
1
)
˜
···
E(
˜
(4.19)
Va
riance
The variance of an individual parameter is given by
∞
−∞
···
∞
E
˜
− µ
x
i
2
x
i
− µ
x
i
2
2
x
i
σ
=
=
f (x
1
,x
2
,...,x
n
) dx
1
···
x
i
dx
n
−∞
(4.20)
[10
Co
variance
For multivariate distributions, another quantity called the covariance
be
comes important. The covariance describes the statistical relationship between two
ra
ndom variables. The covariance is
Lin
—
*
2
——
Lon
PgE
E
x
i
− µ
x
i
x
j
− µ
x
j
σ
x
i
,x
j
=
∞
−∞
···
∞
(4.21)
x
i
− µ
x
i
x
j
− µ
x
j
f (x
1
,x
2
,...,x
n
) dx
1
···
=
dx
n
−∞
W
hereas the variance is always larger than or equal to zero, the covariance can be
ne
gative, positive, or even zero.
[10
Co
rrelation Coefficients
The correlation coefficient of two random variables is
de
fined as
E
˜
x
j
− µ
x
j
x
i
− µ
x
i
˜
σ
x
i
,x
j
σ
ρ
x
i
,x
j
=
=
(4.22)
σ
σ
σ
x
i
x
j
x
i
x
j
Therefore, the correlation coefficient equals the covariance divided by the respective
standard deviations. An important property of the correlation coefficient is that
−
1
≤ ρ
x
i
,x
j
≤
1
(4.23)
If two random variables are stochastically independent, then the covariance (and thus
the correlation coefficient) is zero. By making use of (4.17) for the density function
of stochastically independent random variables, we can write (4.21) as
∞
∞
x
i
− µ
x
i
x
j
− µ
x
j
g
i
(x
i
)g
j
(x
j
)dx
i
dx
j
σ
=
x
i
,x
j
−∞
−∞
∞
x
i
− µ
x
i
g
i
(x
i
)dx
i
∞
−∞
x
j
− µ
x
j
g
j
(x
j
)dx
j
=
(4.24)
−∞
