208
Theorem 1
Let X
*
(r)
be the rth largest prospect potential of a play with a conditional
pool-size distribution H(x) and number-of-pools distribution P (N = n).
For k ≥ 1, let x k < · · · < x 1 denote a sequence of known pool sizes and
let r 1 < r 2 · · · < r k denote the ranks among all pools, both discovered and
undiscovered, of the given pool sizes. The conditional density of X
*
(r)
,
given that X
*
(r 1)
= x 1 , ... , X
*
(r k)
= x k and X
*
(r)
> 0, denoted by f (x|x 1 , ... , x k ), is
the following:
For
1.
r j < r < r j+1 and x j+1 < x < x j ,
(
)
(
)
(
) ( )
( ) ( )
( )
( )
( ) ( )
( )
G
G
G
+
+
+
+
− −
− −
+
− −
+
−
=
−
−




−
−




×


−


!
1
1
1
1
1
1
1
1
1
, ,
h
j
j
j
j
j
j
k
j
j
r
r
r r
j
j
r
r r
j
j
r
r
f x x
x
r
r
r r
H x
H x
H x
H x
x
H x
H x
(D.1)
where G (m) = (m 2 1) !, h (x) is the probability density function of a pool size,
and ( )
h( )
1
∞
=
=
∫ x
H x
z d z
minus the cumulative distribution function.
Appendix D: Pool Size Conditional
on Pool Ranks