3. The

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:

, i

=1,2,…,N (1)

where

and , if i > l.

The value Si,j called the ith

binomial moment of µ.

By consideration

b1, b2…,bN

as variables and estimate ;,

then we have the following linear programming problems:

Minimize {b1

+ b2 + …+ bV +…+ bN} (2)

Subject to:

Maximize {b1

+ b2 + …+ bV +…+ bN} (3)

Subject to:

Solving these problems will give us the best possible of lower and upper

bounds for:

(4)

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.

3.1. The

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:

. (5)

Where:

(6)

And the upper bound of by:

(7)

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:

, (8)

where .

5. The Bounds of Increasing MS C(k,r,n:F).

Using

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)

Evaluation

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)

Furthermore,

one value of given by:

. (15)

The maximum error is:

. (16)