The GNSS integer ambiguities: estimation and validation

The expectation, E{x}, and the dispersion, D{x}, of x ∼ χ 2 (n, λ) are:
E{x} = n + λ and D{x} = 2n + 4λ (A.13)
If x ∼ N(µ, Q) and y = x T Q −1 x then:
y ∼ χ 2 (n, λ) with λ = µ T Q −1 µ (A.14)
A.2.3 The F -distribution
A scalar random variable, x, has a non-central F -distribution with m and n degrees of
freedom and non-centrality parameter λ, if its PDF is given as:
⎧
⎨exp{−
fx(x) =
⎩
λ
∞ (
2 }
j=0
λ
2 )jx m 2 +j−1 m m 2 +j n n 2 Γ( m n
2 + 2 +j)
j!Γ( m n
2 +j)Γ( 2 )(n+mx) m 2 + n 2 +j for 0 < x < ∞
(A.15)
0 for x ≤ 0
The following notation is used:
x ∼ F (m, n, λ) (A.16)
If λ = 0, the distribution is referred to as the central F -distribution.
The expectation, E{x}, and the dispersion, D{x}, of x ∼ F (m, n, λ) are:
E{x} = n
n − 2 (n > 2) and D{x} = 2n2 (m + n − 2)
m(n − 2) 2 (n > 4) (A.17)
(n − 4)
If u ∼ N(u, Qu), v ∼ N(v, Qv), and u and v are uncorrelated, then
m×1
n×1
x = uT Q −1
u u/m
v T Q −1
v v/n
is distributed as:
x ∼ F (m, n, λ) with λ = u T Q −1
u u (A.18)
The distribution of x = u T Q −1
u u/m is given as: x ∼ F (m, ∞, λ).
A.2.4 Student’s t-distribution
A scalar random variable, x, has a Student’s t-distribution with n degrees of freedom,
if its PDF is given as:
fx(x) = Γ
n+1
2
√
n nπΓ 2
1 + x2
−
n
n+1
2
for −∞ < x < ∞ (A.19)
Parameter distributions 147