GENERAL BACKGROUND - GPS Satellite Surveying - page 363

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x n ) are n stochastically independent variables, each hav-
ing a normal distribution, with different means

Assume that ( x 1 , ˜ x 2 , ˜ ··· , ˜

2

µ i and variances σ

i . Then the linear

function

y ˜ = k 1 ˜ x 1 + k 2 ˜ x 2 +···+ k n ˜ x n (A.141)

is distributed as

n n

n

k i σ

i

y ˜ ∼ k i µ i , (A.142)

i i

If the random variable w has a standardized normal distribution, i.e., ˜ w ˜ ∼ ( 0 , 1 ) ,

th en the square of the standardized normal distribution

[36

w 2

1

v ˜ =˜ ∼ χ (A.143)

Lin
—
5.0
——
Lon
PgE

ha s a chi-square distribution with one degree of freedom.
Assume that ( x n ) are n stochastically independent random variables,

ea ch having a chi-square distribution. The degrees of freedom r i can differ. Then the
ra ndom variable

x 1 , ˜ x 2 , ˜ ··· , ˜

y ˜ =˜ x 1 +˜ x 2 +···+ ˜ x n (A.144)

is distributed

[36

2
Σ

y ˜ ∼ χ (A.145)

r i

Th e degree of freedom equals the sum of the individual degrees of freedom.
Assume (

˜ ˜ ··· ˜

x n ) are n stochastically independent random variables, each
ha ving a normal distribution. The means are nonzero. Then

x 1 , x 2 , ,

˜

2

n

x i − µ i
σ i

w 2

2
n

y ˜ ∼ ˜ = ∼ χ (A.146)

x n ) are n stochastically independent normal random
variables with different means

Assume that ( x 1 , ˜ x 2 , ˜ ··· , ˜

2

µ i and variances σ

i . Then the sum of squares

x i

2
n, λ

y ˜ = ˜ ∼ χ (A.147)

ha s a noncentral chi-square distribution. The degree of freedom is n and the noncen-
tra lity parameter is

µ

i

λ = (A.148)

i

σ