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.
"Moments of Random Group Size Distributions." Ann. Math. Statist. 20 (4) 523 - 539, December, 1949. https://doi.org/10.1214/aoms/1177729945