Global Positioning System Reference
In-Depth Information
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∂f
1
∂x
1
∂f
1
∂x
2
∂f
1
∂x
u
···
∂f
2
∂x
1
∂f
2
∂x
2
∂f
2
∂x
u
∂
f
∂
x
=
n
G
u
=
···
(A.121)
.
.
.
···
∂f
n
∂x
1
∂f
n
∂x
2
∂f
n
∂x
u
···
The point of expansion is
P(
x
x
0
)
. Every component of
y
is differentiated with
respect to every variable. Thus, the matrix
G
has as many columns as there are
parameters, and as many rows as there are components in
y
. The components of
f
(
x
0
)
ar
e equal to the respective functions evaluated at
x
0
.
=
[35
A
.5 STATISTICS
Lin
—
0.2
——
Nor
PgE
Br
ief explanations are given on one-dimensional distributions and hypothesis testing.
Th
e material of this appendix can be found in the standard literature on statistics. The
ex
pressions for the noncentral distribution are given, e.g., in Koch (1988).
A.5.1 One-Dimensional Distributions
Th
e chi-square density function is given by
[35
/
1
1
(r/
2
)
x
(r/
2
)
−
1
e
−
x/
2
x>
0
2
r/
2
Γ
f(x)
=
(A.122)
2
0
elsewhere
The symbol
r
denotes a positive integer and is called the degree of freedom. The
m
ean, i.e., the expected value, equals
r
, and the variance equals 2
r
. The degree of
fre
edom is sufficient to describe completely the chi-square distribution. The symbol
Γ
denotes the well-known gamma function, which is dealt with in topics on advanced
ca
lculus and can be written as
Γ
(g)
=
(g
−
1
)
!
(A.123)
g
√
π
1
2
2
2
g
Γ
(
2
g)
Γ
Γ
+
=
(A.124)
(g)
for positive integer
g
. Examples of the chi-square distribution for small degrees of
freedom are given in Figure A.2. The probability that the random variable
x
is less
˜
than
w
α
is
