### Abstract

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>We derive the joint distribution of Y_{1},Y_{2},...,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 Y_{1},Y_{2},... are stochastically increasing (respectively, decreasing). In particular, if Y=X^{1/α} where X is uniformly distributed on [0,1] then the significant digits Y_{1},Y_{2},... are stochastically increasing (respectively, decreasing) if α<1 (respectively, α>1).

Original language | English (US) |
---|---|

Pages (from-to) | 263-270 |

Number of pages | 8 |

Journal | Statistics and Probability Letters |

Volume | 46 |

Issue number | 3 |

DOIs | |

Publication status | Published - Feb 1 2000 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Statistics and Probability Letters*,

*46*(3), 263-270. https://doi.org/10.1016/S0167-7152(99)00111-X