
82 Statistical Computing in Nuclear Imaging
Conditioned on the number of decays (y ∈ Y
c
a
nd c ≤ d
1
)
p(y|d) =
I
Y
i=1
c
i
!
(ǫ
i
)
c
i
K
Y
k=1
(α
ki
)
y
ki
y
ki
!
!
d
i
!
c
i
!(d
i
− c
i
)!
(ǫ
i
)
c
i
(1 − ǫ
i
)
d
i
−c
i
=
=
I
Y
i=1
p (y
i
|d
i
) =
I
Y
i=1
p(y
i
|c
i
)p(c
i
|d
i
) = p(y|c)p(c|d) (3.25)
1
c ≤ d ≡ ∀i : c
i
≤ d
i
.
Conditioned on the number of radioactive nuclei (y ∈ Y
c
and c ≤ r
1
)
p(y|r) =
I
Y
i=1
c
i
!
(ǫ
i
)
c
i
K
Y
k=1
(α
ki
)
y
ki
y
ki
!
!
r
i
!
c
i
!(r
i
− c
i
)!
(ǫ
i
q)
c
i
(1 − ǫ
i
q)
r
i
−c
i
=
=
I
Y
i=1
p(y
i
|r
i
) =
I
Y
i=1
p(y
i
|c
i
)p(c
i
|r
i
) = p(y|c)p(c|r) (3.26)
1
c ≤ r ≡ ∀i : c
i
≤ r
i
.
If the conditions given in the parentheses in headers of each conditional
Equations (3.24)–(3 .26) are not met, the distributions ar e zero. The above
equations are the basic ...