Global Positioning System Reference
In-Depth Information
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E(c
x)
˜
=
cE(
x)
˜
(4.29)
The expected value (mean) of a constant equals the constant. Because the mean is a
constant, it follows that
E
E(
x)
= µ
x
˜
(4.30)
Re
lations (4.28) and (4.29) also hold for multivariate density functions, as can be seen
fro
m (4.18). Let
y
˜
=˜
x
1
+˜
x
2
be a linear function of random variables, then
∞
∞
E (
x
1
+˜
˜
x
2
)
=
(x
1
+
x
2
) f (x
1
,x
2
) dx
1
dx
2
−∞
−∞
∞
∞
∞
∞
[10
=
x
1
f (x
1
,x
2
) dx
1
dx
2
+
x
2
f (x
1
,x
2
) dx
1
dx
2
−∞
−∞
−∞
−∞
Lin
—
6.5
——
No
PgE
=
˜
+
˜
E (
x
1
)
E (
x
2
)
(4.31)
Thus, the expected value of the sum of two random variables equals the sum of the
individual expected values. By combining (4.28) and (4.31), we can compute the
expected value of a general linear function of random variables. Thus, if the elements
of the
n
×
×
u
matrix
A
and the
n
1 vector
a
0
are constants and
y
=
a
0
+
Ax
(4.32)
[10
then the expected value is
E (
y
)
=
a
0
+
A
E (
x
)
(4.33)
Th
is is the law for propagating the mean. The law of variance-covariance propagation
is
as follows:
E
y
−
µ
y
T
−
µ
y
y
Σ
y
≡
E
y
E(
y
)
T
E(
y
)
y
=
−
−
E
y
A
E(
x
)
T
A
E(
x
)
y
=
−
a
0
−
−
a
0
−
(4.34)
E
[
Ax
A
E(
x
)
]
T
=
−
A
E(
x
)
][
Ax
−
A
E
[
x
E(
x
)
]
T
A
T
=
−
E(
x
)
][
x
−
Σ
x
A
T
=
A
The first line in Expression (4.34) is the general expression for the variance-covariance
matrix of the random variable
y
according to definition (4.26);
µ
y
is the expected
