78
7 Regression Models and Hypothesis Testing
where we introduce spherical coordinates with r
2
= z
2
1 + z
2
2 + · · · z
2
n in the second
equation. Since the integrand does not depend on angular variables we know that
d = S n is the surface area of an n−dimensional sphere. After substituting q = r
2
we arrive at
1 =
S n
2(2π) n/2
∞
0
q
n/2−1 e
−q/2 dq =
∞
0
ψ n (q)dq ,
(7.18)
which implicitely defines the probability distribution ψ n (q) for the q = χ
2
.
The previous equation still depends on the unknown surface area S n . We can, however, determine S n from the requirement that the distribution function is normalized.
Substituting p = q/2 we find
1 =
S n
2(2π) n/2 2
n/2−1
∞
0
dpp
n/2−1 e
− p
=
S n
2(2π) n/2 2
n/2−1
(n/2)
(7.19)
where (z) is the Gamma function [4], defined as
(z) =
∞
0
t
z−1 e
−t dt .
(7.20)
Solving for S n , we find
S n =
2π
n/2
(n/2)
,
(7.21)
which we insert into (7.18) to obtain the following expression for the probability
distribution function
ψ n (q)dq =
1
2 n/2 (n/2)
q
n/2−1 e
−q/2 dq .
(7.22)
Figure 7.4 displays ψ n (q) for n = 2, 5, and 10.
The probability of finding a χ
2 smaller than a given limit ξ is given by the integral
from zero to ξ of the probability distribution function
Q n (ξ ) =
ξ
0
ψ n (q)dq =
ξ
0
1
2 n/2 (n/2)
q
n/2−1 e
−q/2 dq = P(n/2, ξ/2) (7.23)
where P(a, x) is the incomplete gamma function [4]. Now we are in a position to
answer the question whether a χ
2 that arises from a fitting procedure or regression
analysis is actually probable. In particular, the probability of finding a value of χ
2
7 Regression Models and Hypothesis Testing
where we introduce spherical coordinates with r
2
= z
2
1 + z
2
2 + · · · z
2
n in the second
equation. Since the integrand does not depend on angular variables we know that
d = S n is the surface area of an n−dimensional sphere. After substituting q = r
2
we arrive at
1 =
S n
2(2π) n/2
∞
0
q
n/2−1 e
−q/2 dq =
∞
0
ψ n (q)dq ,
(7.18)
which implicitely defines the probability distribution ψ n (q) for the q = χ
2
.
The previous equation still depends on the unknown surface area S n . We can, however, determine S n from the requirement that the distribution function is normalized.
Substituting p = q/2 we find
1 =
S n
2(2π) n/2 2
n/2−1
∞
0
dpp
n/2−1 e
− p
=
S n
2(2π) n/2 2
n/2−1
(n/2)
(7.19)
where (z) is the Gamma function [4], defined as
(z) =
∞
0
t
z−1 e
−t dt .
(7.20)
Solving for S n , we find
S n =
2π
n/2
(n/2)
,
(7.21)
which we insert into (7.18) to obtain the following expression for the probability
distribution function
ψ n (q)dq =
1
2 n/2 (n/2)
q
n/2−1 e
−q/2 dq .
(7.22)
Figure 7.4 displays ψ n (q) for n = 2, 5, and 10.
The probability of finding a χ
2 smaller than a given limit ξ is given by the integral
from zero to ξ of the probability distribution function
Q n (ξ ) =
ξ
0
ψ n (q)dq =
ξ
0
1
2 n/2 (n/2)
q
n/2−1 e
−q/2 dq = P(n/2, ξ/2) (7.23)
where P(a, x) is the incomplete gamma function [4]. Now we are in a position to
answer the question whether a χ
2 that arises from a fitting procedure or regression
analysis is actually probable. In particular, the probability of finding a value of χ
2