206
Appendix C
=
−
−
−
=
+
∑ 1
1
1
2
ln( )
( )
p
j
j
p
p
x c
x
m
m
b
m
s
(C.12)
and variance matrix
2
1
1
(
)
T
p
p
−
−
=
−
V
b b s
⌺
(C.13)
Note that V p – 1 is independent of the given pool size x.
Let
1
1
1
( )
( )
( )
p
p
m x
x
m x
−
−
=
m
#
−
−
−
− −
=
1,1
1, 1
1
1,1
1, 1
...
...
p
p
p
p p
v
v
v
v
V
Then, for i 5 1, 2, ..., p–1, we have
1
( )
2
E
i
i i
m x
v
i
Z X x e
+
= =
(C.14)
and
1
2 ( )
(
1)
2
Var
ii
i
i i
m x
v e
i
Z X x e
+
−
= =
(C.15)
The conditional distribution of Z p , given that X 5 x, is lognormal with
parameters
( )
−
−
−
−
=
− ∑
1
1
1
1
1
ln
( ) and
p
T
j
p
p
p
j
x c
m x
a
a
V
(C.16)
Appendix C
=
−
−
−
=
+
∑ 1
1
1
2
ln( )
( )
p
j
j
p
p
x c
x
m
m
b
m
s
(C.12)
and variance matrix
2
1
1
(
)
T
p
p
−
−
=
−
V
b b s
⌺
(C.13)
Note that V p – 1 is independent of the given pool size x.
Let
1
1
1
( )
( )
( )
p
p
m x
x
m x
−
−
=
m
#
−
−
−
− −
=
1,1
1, 1
1
1,1
1, 1
...
...
p
p
p
p p
v
v
v
v
V
Then, for i 5 1, 2, ..., p–1, we have
1
( )
2
E
i
i i
m x
v
i
Z X x e
+
= =
(C.14)
and
1
2 ( )
(
1)
2
Var
ii
i
i i
m x
v e
i
Z X x e
+
−
= =
(C.15)
The conditional distribution of Z p , given that X 5 x, is lognormal with
parameters
( )
−
−
−
−
=
− ∑
1
1
1
1
1
ln
( ) and
p
T
j
p
p
p
j
x c
m x
a
a
V
(C.16)