Moments of Random Group Size Distributions

Abstract

A number of practical problems involve the solution of a mathematical problem of the class described in the classical language of probability theory as follows: "A number of balls are independently distributed among a number of boxes, how many boxes contain no balls, 1 ball, 2 balls, 3 balls, and so on." Problems arising in the oxidation of rubber and the genetics of bacteria are discussed as applications. A method is given of solving problems of this sort when "how many" is adequately answered by the calculation of means, variances, covariances, third moments, etc. The method is applied to a number of the simplest cases, where the number of balls is fixed, binomially distributed or Poisson and where the "sizes" of the boxes are equal or unequal.