Upper and lower bounds for the probability that r events occur

Abstract

Upper and lower bounds are derived for the probability that at least r events occur and exactly r events occur. It is assumed that the probabilities of intersection of at most m events are known. The method of indicator random variables leads to a linear programming problem which is transformed into a problem of determining the coefficients of an mth degree polynomial. The derived bounds are better than the Bonferroni and Galambos bounds, also Jordan’s formulae follow as a special case. Specific applications are presented indicating the supperiority of the new bound.

Keywords:

• Indicator variables
• Polynomial
• Maximization
• Minimization
• Optimal