Q2 Z2,(Q2) Z2(Q2) - Institute for Water Resources - U.S. Army

(34) P 2 (X)
(36)
1
=
1 a
b c '
and Pl(X) -
2 a k • b
a a
•
b b
The two formulas give different results for the same probability. P 2 (X)
should be taken as the correct result, since it is based on the full
available information concerning mode 2. It remains that one cannot be
sure that the sum of these P (X)'s equals 1. In general, they will not add
to one, but the divergence should be small. Still using (30) 4
.
(35)
3 .
E p (x) • 1 2 •■■■,...... .1. 1 .........■ f
i
+ 3
ak
+ 3
bb
1 + -a- + -; l +
b a.
r + 3
Assuming k/b > c/b, the second term of (32) is smaller than the second term
of (28). It is exactly smaller by
be •
K - (a + b + c) (A b + c + e)
, if k = c + e.
Similarly, if k/b c/b the third term of (32) is greater by
(37) L
ce
(a + b + c) (a + b + c e) 9
if k
c a
K and L should be small, and as they tend to compensate each other, the
sumoftheP. X) should be close to one. We conclude that the use of
1 (
P (X) would be reasonably safe.
. •
Obviously, no such problem arises in the computation of
1 1
(38) P..(X.
13 ij
) =
- (A' ' X )
li li 1 + e ji + b ji
1 +
Pij (X )
il
;2