The object of this note is to suggest a method, believed by the writer to be new, whereby once a pension fund contribution has been calculated the revised contribution occasioned by a small change in the rate of interest can be approximately determined with great ease and rapidity.

Let 100cx be the percentage contribution for entry age x required to support a pension per annum equal to 100k % of pensionable salary for each year of service. Assume for simplicity that normal retirements and early retirements (if any) can together be sufficiently allowed for by the use of a single average retirement age (M).

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.

In his paper (J.S.S. Vol. VI, p. 172) Dr Wishart showed that the variance-ratio test can be very easily used with the aid of the appropriate tables. It may not have been obvious to the reader of that paper that, unlike most distribution functions, the one in question can be evaluated by elementary methods and that a simple formula can be established to give the required probability independently of any table. It is not suggested that the use of the formula to be demonstrated in this note is preferable to the ordinary employment of tables. It may be remarked, however, that in certain cases where a probability has to be directly calculated the formula has a big advantage, since the tables give only the values of F (or z) corresponding to a few specific ‘probability points’ and interpolation between these must be very unsatisfactory, as is obvious from a glance at Table 3 of Dr Wishart's paper.