Global Positioning System Reference
In-Depth Information
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˜
w
˜
˜
=
t
(A.131)
v/r
has a
t
distribution with
r
degrees of freedom. The distribution function is
1
−
(r
+
1
)/
2
t
2
r
=
Γ
[
(r
+
1
)/
2]
f(t
r
)
√
π
+
−∞
<t<
∞
(A.132)
r
Γ
(r/
2
)
The density function (A.132) is symmetric with respect to
t
=
0. See Figure A.4.
Furthermore, if
r
=∞
then the
t
distribution is identical to the standardized normal
distribution; i.e.,
t
∞
=
n(
0
,
1
)
(A.133)
[36
The density in the vicinity of the mean (zero) is smaller than for the unit normal
distribution, whereas the reverse is true at the extremities of the distribution. The
t
distribution converges rapidly toward the normal distribution. If the random variable
˜
Lin
—
*
2
——
No
PgE
∼
δ
w
,
1
)
is normal distributed with unit variance but with a nonzero mean, then
th
e function (A.131) has a noncentral
t
distribution with
r
degrees of freedom and a
no
ncentrality parameter
n(
.
Consider two stochastically independent random variables,
δ
r
2
,
di
stributed with
r
1
and
r
2
degrees of freedom, respectively; then the random variable
r
1
u
˜
∼ χ
and
v
˜
∼ χ
u/r
1
˜
˜
F
=
(A.134)
[36
v/r
2
has the density function
r
2
)/
2]
(r
1
/r
2
)
r
1
/
2
F
(r
1
/
2
)
−
1
=
Γ
[
(r
1
+
f(
F
r
1
,r
2
)
0
<F <
∞
(A.135)
Γ
(r
1
/
2
)
Γ
(r
2
/
2
)
r
1
F/r
2
)
(r
1
+
r
2
)/
2
(
1
+
Figure A.4
The probability density function of the
t
distribution.
