Global Positioning System Reference
In-Depth Information
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Mean
The mean, also called the expected value of a continuously distributed ran-
dom variable, is defined as
∞
µ
x
=
E(
x)
˜
=
x f (x) dx
(4.11)
−∞
Th
e mean is a function of the density function of the random variable. The integration
is
extended over the whole population. Equation (4.11) is the analogy to the weighted
m
ean in the case of discrete distributions.
Variance
The variance is defined by
∞
E
˜
− µ
x
2
x
− µ
x
2
f(x)dx
x
σ
=
x
=
(4.12)
[10
−∞
The variance measures the spread of the probability density in the sense that it
gives the expected value of the squared deviations from the mean. A small variance
therefore indicates that most of the probability density is located around the mean.
Lin
—
*
2
——
Lon
*PgE
Multivariate Distribution
Any function
f(x
1
,x
2
,...,x
n
)
of
n
continuous vari-
ables
˜
x
i
can be a joint density function provided that
f (x
1
,x
2
,...,x
n
)
≥
0
(4.13)
∞
−∞
···
∞
[10
f (x
1
,x
2
,...,x
n
) dx
1
···
dx
n
=
1
(4.14)
−∞
It follows as a natural extension from (4.10) that
a
1
−∞
···
a
n
P (
x
1
<a
1
,...,
˜
x
n
<a
n
)
˜
=
f (x
1
,x
2
,...,x
n
) dx
1
···
dx
n
(4.15)
−∞
The marginal density of a subset of random variables
(x
1
,x
2
,...,x
p
)
is
∞
−∞
···
∞
g
x
1
,x
2
,...,x
p
=
f (x
1
,x
2
,...,x
n
) dx
p
+
1
dx
p
+
2
···
dx
n
(4.16)
−∞
Stochastic Independence
The concept of stochastic independence is required
when dealing with multivariate distributions. Two sets of random variables,
(
x
1
,...,
˜
x
p
)
and
(
x
n
)
, are stochastically independent if the joint density function
can be written as a product of the two respective marginal density functions, e.g.,
˜
x
p
+
1
,...,
˜
˜
g
1
x
1
,x
2
,...,x
p
g
2
x
p
+
1
,x
p
+
2
,...,x
n
f (x
1
,x
2
,...,x
n
)
=
(4.17)
Vector of Means
The expected value for the individual parameter
x
i
is
