Radix expansions and the uniform distribution

Rameshwar D. Gupta and Donald St. P. Richards

Statistics & Probability Letters, 2000, vol. 46, issue 3, 263-270

Abstract: Let the random variable X be uniformly distributed on [0,1], [alpha] be a positive number, [alpha][not equal to]1, and b be a positive integer, b>1. We derive the joint distribution of Y1,Y2,...,Yk, the first k significant digits in the radix expansion in base b of Y=X1/[alpha]. We show that, as k-->[infinity], Yk converges in distribution to the uniform distribution on the set {0,1,...,b-1}. We also prove that if Y is a random variable taking values in [0,1] whose cumulative distribution function is continuous and convex (respectively, concave) then the significant digits Y1,Y2,... are stochastically increasing (respectively, decreasing). In particular, if Y=X1/[alpha] where X is uniformly distributed on [0,1] then the significant digits Y1,Y2,... are stochastically increasing (respectively, decreasing) if [alpha] 1).

Keywords: Beta; distribution; Power; transformation; Random; number; generation; Significant; digits; Stochastic; ordering; Uniform; distribution (search for similar items in EconPapers)
Date: 2000
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