How to Make a Lumpy Random-Number Generator

Normally, random-number generators are designed to produce numbers with a uniform distribution. The sum of uniform random variates has a bell-curveshaped distribution. Using bell curves like wavelets, individual uniform random variables can be summed to produce arbitrary nonuniform distributions. The result is a simple, customizable non-uniform random-number generating algorithm that has been prototyped on Plan 9 and is equally suitable for other computing environments, including very small embedded systems. 1. Problem definition Normally, random-number generators produce uniformly distributed values. However, nonuniform random numbers are needed for a number of purposes. In simulation, one often needs random numbers conforming exactly to the observed or theoretical distribution of an input variable, in order to produce an authentic distribution of simulated output [1]. In other situations, the requirements are much less precise, but random numbers with a preference for certain values or ranges are still desired. Examples include equalizing wear on machinery by having a machine return to approximately but not exactly the same position each time; introducing “dither” to avoid unwanted synchronism with external processes; and correcting for nonuniformity in some process downstream from the random number generator. Traditionally, nonuniform distributions are generated by transforming the output of a uniform random-number generator, often using elaborate floating-point computations. In this paper I outline an alternative that is especially suitable for the latter set of cases, where quick computation is more important than hitting a specified distribution exactly.1 2. Approach As Fig. 1 shows, the sum of n uniform random variables is an (n−1)th-degree polynomial approximation to a normal distribution (bell curve) ([2], p. 22). For practical purposes, the curve with n=3 is smooth enough. Thus, one can generate a bell curve distribution by merely generating three random numbers each time, adding them, and dividing by three. Bell curves can be used like wavelets to synthesize more complex curves. For example, the distribution in Fig. 2 was synthesized from a nonzero baseline mixed with three bell curves by the code in Fig. 5. This research was done for CORAID, Inc. (www.coraid.com) while the author was on summer leave from the Institute for Artificial Intelligence, University of Georgia (www.ai.uga.edu). The author thanks Brantley Coile for posing the initial question from which this project arose. Figure 1: Histograms of the sum of n uniform random variables, from 10 000 000 trials.