Pool Size Conditional on Pool Ranks
211
The expectation of the ratio is given by
( )
(
) ( )
(
)( )
(
)
(
)
1
2
2
2
,
2
1
1
1
1
2 1
2
0
2
ERPS
P
,
1
P
r r
n r
r r
r
r r
r
N n
J u n r
u
du
N r
`
G
G
G
=
− −
=
−
=
×
−
∫
≥
∑
(D.12)
where
(
)
( ) ( )
( )
( )
2
2 1
1
2
2
0
2
1
,
1
n r
r
n
J u n r
r
H uH x H x
H x
h x dx
r
x
`
−
−
−
=
−
∫
(D.13)
For proofs of Theorem 1 and its Corollary and Theorem 2, see Lee
and Wang (1986) and refer to Appendix A.
211
The expectation of the ratio is given by
( )
(
) ( )
(
)( )
(
)
(
)
1
2
2
2
,
2
1
1
1
1
2 1
2
0
2
ERPS
P
,
1
P
r r
n r
r r
r
r r
r
N n
J u n r
u
du
N r
`
G
G
G
=
− −
=
−
=
×
−
∫
≥
∑
(D.12)
where
(
)
( ) ( )
( )
( )
2
2 1
1
2
2
0
2
1
,
1
n r
r
n
J u n r
r
H uH x H x
H x
h x dx
r
x
`
−
−
−
=
−
∫
(D.13)
For proofs of Theorem 1 and its Corollary and Theorem 2, see Lee
and Wang (1986) and refer to Appendix A.