Exact probabilistic and mathematical proofs of the relation between the mean-mu and the generalized 80/20-rule

Abstract

The generalized 80/20-rule states that 100 x % of the most productive sources in (for example) a bibliography produce 100 y % of the items and one is interested in the relation between y and x. The following (intuitively clear) property (*) is investigated: suppose we have two bibliographies with average number of items per source mu1, mu2, respectively, such that mu1 < mu2. Then y1 = y2 implies x1 > x2, i.e., in the second bibliography we need a smaller fraction of the most productive sources than in the first one in order to have the same fraction of the items produced by these sources. First, we prove this theorem in a probabilistic way for the geometric distribution and for the Lotka distribution with power alpha > 2. A remarkable result is found, when allowing for zero-item sources in case of the geometric distribution: the opposite property (*) is found to be true (x1 < x2 instead Of x1 > x2). Then a mathematical proof of property follows for Lotka's function with powers alpha = 0, alpha = 1, alpha = 1.5, alpha = 2, where the problem remains open for the other alpha's. The difference between both approaches lies in the fact that the probabilistic proofs use distributions with arguments until infinity, while the mathematical proofs use exact functions with finite arguments.