3. b2…,bN as variables and estimate ;,

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.

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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)

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