Exact probabilistic and mathematical proofs of the relation between the mean μ and the generalized 80/20‐rule

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 μ1, μ2, respectively, such that μ1 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 α x2 instead of x1 > x2). Then a mathematical proof of property (*) follows for Lotka's function with powers α = 0, α = 1, α = 1.5, α = 2, where the problem remains open for the other α'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. © 1993 John Wiley & Sons, Inc.