Uniform occurrence of digits for folded and mixture distributions on finite intervals

Abstract

Historically, the occurrences of digits in a set of observed values of a random variable are of interest in theoretical and applied statistics for various reasons including rounding errors and statistical forensics. A probabilistic approach to such occurrences is needed in order to draw conclusions about the phenomena represented by these observed values. In this paper, we study certain structural properties of finite decimal expansions of real numbers from finite intervals within the framework of 10 k -folded distributions defined on such intervals. To provide a probabilistic structure for the occurrence of digits we obtain characterization results for the rectangular and folded-rectangular distributions. A brief discussion of the use of 10 k -bin histogram distributions to determine the approximate distribution of digits, along with two examples, illustrate our result.