It is shown that the distribution of the number of ones in the binary expansion of an integer chosen uniformly at random from the set 0, 1,…, n − 1 can be approximated in total variation by a mixture of two neighbouring binomial distributions, with error of order (log n)−1. The proof uses Stein's method.

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Barbour, A D (1992). *On the binary expansion of a random integer.* Statistics and Probability Letters, 14(3):235-241.

## Abstract

It is shown that the distribution of the number of ones in the binary expansion of an integer chosen uniformly at random from the set 0, 1,…, n − 1 can be approximated in total variation by a mixture of two neighbouring binomial distributions, with error of order (log n)−1. The proof uses Stein's method.

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## Additional indexing

Item Type: | Journal Article, refereed, original work |
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Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Uncontrolled Keywords: | Random integer; binary expansion; Stein's method; binomial mixtures |

Language: | English |

Date: | 1992 |

Deposited On: | 12 Apr 2010 15:06 |

Last Modified: | 05 Apr 2016 13:28 |

Publisher: | Elsevier |

ISSN: | 0167-7152 |

Free access at: | Related URL. An embargo period may apply. |

Publisher DOI: | 10.1016/0167-7152(92)90028-4 |

Related URLs: | http://user.math.uzh.ch/barbour/pub/Barbour/BChen_Binary.pdf (Author) |

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