A tutorial on how to efficiently calculate and format tables of the binomial distribution

الملخص EN

Owing to the importance of the binomial distribution, tables for its probability mass function (PMF) and its cumulative distribution function (CDF) have been extensively published.

This paper presents a tutorial on a novel technique for constructing and recording these tables.

The tables are to be compiled with a high precision and a minimal round-off error through the use of highly efficient iterative algorithms based on binary recursive relations for the PMF and the CDF.

The computational cost for each additional table entry is merely a single multiplication plus two additions.

This remarkably low cost is due to the fact that each entry is computed in terms of exactly two preceding entries.

Certain symmetries and limiting values are noted, thereby allowing a considerable reduction in the number of listed entries.

Table entries are given in compact forms of fixed-point or floating-point format, with an objective of minimizing the numerical uncertainty that arises when representing an entry in a limited field width, i.e., when using a limited number of digit or character positions.

This kind of novel format is crucial for accurate evaluation of probabilities associated with ultra-high reliability systems or with very rare events.