Radix expansions and the uniform distribution

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• 作者：Gupta ; Rameshwar D. ; et. al.
• 关键词：Beta distribution ; Power transformation ; Random number generation ; Significant digits ; Stochastic ordering ; Uniform distribution
• 刊名：Statistics and Probability Letters
• 出版年：2000
• 期刊代码：56_01677152
• 类别：mt
• 出版时间：February 1, 2000
• 卷：46
• 期：3
• 页码：263-270
• 文件大小：98.7 K

摘要

Let the random variable X be uniformly distributed on [0,1], α be a positive number, α≠1, and b be a positive integer, b>1. We derive the joint distribution of Y₁,Y₂,…,Y_k, the first k significant digits in the radix expansion in base b of Y=X^1/α. We show that, as k→∞, Y_k 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=X^1/α where X is uniformly distributed on [0,1] then the significant digits Y₁,Y₂,… are stochastically increasing (respectively, decreasing) if α<1 (respectively, α>1).