The GNSS integer ambiguities: estimation and validation

Let x be normally distributed with expectation and dispersion as in (A.4), and let y
be defined as the m × 1 random vector given by y = Ax + a. Then according to the
propagation law of the mean and of variances:
E{y} = Aµ + a and D{x} = AQA T
and thus:
(A.5)
y ∼ N(Aµ + a, AQA T ) (A.6)
The cumulative normal distribution is given by:
Φ(x) =
x
−∞
Some useful remarks:
y
x
1
√ exp{−
2π 1
2 v2 }dv (A.7)
1
√ 2π exp{− 1
2 v2 }dv = Φ(y) − Φ(x) (A.8)
Φ(−x) = 1 − Φ(x) (A.9)
A.2.2 The χ 2 -distribution
A scalar random variable, x, has a non-central χ 2 -distribution with 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 n 2 +j−1 exp{− x
2 }
j!2 n 2 +j Γ( n
2 +j)
for 0 < x < ∞
0 for x ≤ 0
with the gamma function:
(A.10)
∞
Γ(x) = t x−1 exp{−t}dt, x > 0 (A.11)
0
The values of Γ(x) can be determined using Γ(x + 1) = xΓ(x), with Γ( 1
2 ) = √ π and
Γ(1) = 1.
The following notation is used:
x ∼ χ 2 (n, λ) (A.12)
If λ = 0, the distribution is referred to as the central χ 2 -distribution.
146 Mathematics and statistics