SELECTIVE INCLUSION-EXCLUSION METHODS AND APPLICATIONS

This thesis introduces new methods of obtaining upper and lower bounds for probabilities of unions and other values which otherwise may be impossible to calculate exactly. These methods, called selective inclusion-exclusion procedures, are derived from the general inclusion-exclusion formulas proposed by Boole (1854) and Bonferroni (1936).; Selective inclusion-exclusion procedures, when applied to obtaining upper/lower bounds for probabilities of unions, allow one to subtract or add only some of the probabilities of k-way intersections of events, as opposed to the general inclusion-exclusion bounds which require addition or subtraction of probabilities of all k-way intersections of events. Because of this, selective inclusion-exclusion procedures can produce tighter upper and lower bounds than are possible using general inclusion-exclusion models.; These new procedures are applied to simultaneous confidence intervals involving multivariate normal variables. Tables showing the effectiveness of this application are given. Specific multivariate normal probability applications are discussed. Comparisons are made with other methods which obtain upper/lower bounds for probabilities of unions.; The selective inclusion-exclusion methods are extended to provide procedures giving upper/lower bounds for expected values of (i) maximums of random variables, (ii) minimums of random variables, and (iii) Minkowski norms of random variables. In addition, selective inclusion-exclusion methods are modified to produce Borel-Cantelli like theorems giving upper/lower bounds for the probability that an infinite dependent sequence occurs infinitely often.