398 Statistics of Medical Imaging
12.4 Appendices
12.4.1 Appendix 12A
This appendix derives pdf of Z, Eq. (1 2.16), and the probabilities of the over
and under-de tec tion of the number of image regions, P
ov
Eq. (12.18) and P
ud
Eq. (12.19).
A.1 pdf of Z
This section proves Eq. (12.16).
Proof.
Y
K
of Eq. (12.11) has a χ
2
distribution with a degree o f freedom (J − K).
Its pdf is
p
Y
K
(y) =
1
2
J−K
2
Γ(
J−K
2
)
e
−
y
2
y
J−K
2
−1
(y
> 0)
0 (y ≤ 0).
(12.54)
Let Z
K
0
=
1
K
0
ˆσ
0
Y
K
0
a
nd Z
K
1
=
1
K
1
ˆσ
1
Y
K
1
.
Their pdfs are
p
Z
K
(z) =
(
K
0
ˆσ
0
2
)
J−K
0
2
Γ(
J−K
0
2
)
e
−
K
0
ˆσ
0
2
z
z
J−K
0
2
−1
(z
> 0)
0 (z ≤ 0),
(12.55)
p
Z
K
(z) =
(
K
1
ˆσ
1
2
)
J−K
1
2
Γ(
J−K
1
2
)
e
−
K
1
ˆσ
1
2
z
z
J−K
1
2
−1
(z
> 0)
0 (z ≤ 0).
(12.56)
Assume that Z
K
0
and Z
K
1
are indepe ndent. The pdf h(z) of Z = Z
K
1
−Z
K
0
is
h(z) =
Z
∞
−∞
p
Z
K
1
(u)p
Z
K
0
(u − z)du
=
R
∞
0
p
Z
K
1
(u)p
Z
K
0
(u − z)du (z < 0)
R
∞
z
p
Z
K
1
(u)p
Z
K
0
(u − z)du (z ≥ 0).
(12.57)
Substituting Eqs. (12.55), (12.56) into Eq. (12.57), using formula [44]
Z
∞
0
x
n
e
−ax
dx =
n!
a
n+1
(a > 0) , (12.58)
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