46
Statistical Methods for Estimating Petroleum Resources
distribution and (2) the number of pools, N, in the play, or their distribution. The superpopulation concept is assumed for this estimation.
Furthermore, the pool-size distribution and the number-of-pools distribution can vary independently, and can be any type of probability distribution (Fig. 3.10).
If N = 1 (i.e., a single pool play), then the distribution of the largest
and smallest pools is precisely given by the pool-size distribution. More
generally, if X 1 , X 2 , … , X N are pool sizes generated independently from
an identical pool-size distribution denoted by F θ , where u = (μ, s
2
), then
the greater-than distribution of the largest pool among N pools (Lee
and Wang, 1983b) is as follows:
( )
( )
, 1
1 1
for
0
N
N
L x
F x
X
= − −
>




u
(3.10)
The greater-than distribution of the rth largest pool is given by
( )
( )
( )
,
1
for
0,
1,2, ... ,
N
N K
N r
k r
N
L
x
F x
F x
X
r
N
K
−
=
 
=
−
>
=


 


 
∑
u
u
(3.11)
Figure 3.10. Diagram showing the concept of pool-size-by-rank by order
statistics.