The purpose of this note is to establish, by finite difference methods, some of the properties of two probability distributions: (1) the distribution of the sum of n independent variates, for each of which every value in the interval (0, 1) is equally probable; and (2) the distribution of the sum of n independent variates, each of which must take one of f equally probable values, 0, 1, 2, …, f–1. For brevity we shall refer to these distributions as (1) the continuous case and (2) the discrete case. The continuous case is important because it gives a solution to a problem that often arises in practical computing: ‘If n numbers, which are all “rounded-off’ at the same [decimal] place, are added together, what is the chance that the total will be the correct (rounded-off) total, or will be 1, 2, etc., different in the last place from this correct (rounded-off) total?’ (Forster, 1947).
Laplace (1776) gave a proof of the continuous case; and later (1812) gave a proof of the discrete case, from which, by a limiting process, he derived the result for the continuous case. The result for the discrete case had, however, been published much earlier by de Moivre (1711), and proofs had been published by de Montmort (1713) and de Moivre (1730). Laplace's proofs are long and difficult to follow; but, as far as I can discover, no later alternative proofs were put forward until Rietz (1924) gave a much simpler proof of the continuous case. Irwin (1927) used characteristic functions to prove the continuous case; and Hall (1927) gave a remarkable geometrical proof of the continuous case and derived a general expression for its moments.