Bounds of Boole-Bonferroni
The Boole-Bonferroni bounds used for estimate the probability of the
union of the N dependent events. These bounds were derived by Boros and Prékopa
2, 17, and improved by many papers as 2-5, 11, 12, 17. The technique of
these bounds based on solving the linear programming problem.
The following expected value proved by Prékopa 18:
and , if i > l.
The value Si,j called the ith
binomial moment of µ.
as variables and estimate ;,
then we have the following linear programming problems:
+ b2 + …+ bV +…+ bN} (2)
+ b2 + …+ bV +…+ bN} (3)
Solving these problems will give us the best possible of lower and upper
These bounds are named Boole-Bonferroni bounds. When V=2, the solutions of problems
(2) and (3) give us the second order of Boole-Bonferroni bounds as follows.
Bounds of Boole-Bonferroni with V=2:
When V=2 and the binomial moments and,
j = 1, 2, 3,…, H, are evaluated, we can
get the lower bound of by:
And the upper bound of by:
4. The Upper bound of Hunter-Worsley.
Also, the Hunter–Worsley upper bound used for estimate the probability
of the union of the N dependent events. Hunter and Worsley 19,20 derived
an effective upper bound using the binomial moment and some of the specific probabilities
involved in .
This upper bound can be calculated quickly and always sharper than the Boole-Bonferroni
upper bound in (7). The upper bound of Hunter–Worsley is given by:
5. The Bounds of Increasing MS C(k,r,n:F).
the definition of , , the state of increasing MS C(k,r,n:F) is less than j, if at
least one event , , occurred. Then
for all (12)
exactly is very difficult, so we will suggest an
approximation for bounds
of increasing MS C(k,r,n:F) using
Boole-Bonferroni bounds, which given by (5), (7), and using Hunter–Worsley upper bound, which given by (8) . For evaluation these bounds, we need calculation
the binomial moments and , which are the main problem. In the following section, we
proposed new algorithms for calculation the binomial moments and . After estimate the
lower bounds and upper bounds of , one can get
the lower bounds and upper
bounds of by:
LBj = 1- (upper bound of ) , (13)
UBj = 1- (lower bound of ). (14)
one value of given by:
The maximum error is: