### Abstract

For fixed p (0 ≤ p ≤ 1), let {L_{0}, R_{0}} = {0, 1} and X_{1} be a uniform random variable over {L_{0}, R_{0}}. With probability p let {L_{1}, R_{1}} = {L_{0}, X_{1}} or = {X_{1}, R_{0}} according as X_{1} ≥ 1 2(L_{0} + R_{0}) or < 1 2(L_{0} + R_{0}); with probability 1 - p let {L_{1}, R_{1}} = {X_{1}, R_{0}} or = {L_{0}, X_{1}} according as X_{1} ≥ 1 2(L_{0} + R_{0}) or < 1 2(L_{0} + R_{0}), and let X_{2} be a uniform random variable over {L_{1}, R_{1}}. For n ≥ 2, with probability p let {L_{n}, R_{n}} = {L_{n - 1}, X_{n}} or = {X_{n}, R_{n - 1}} according as X_{n} ≥ 1 2(L_{n - 1} + R_{n - 1}) or < 1 2(L_{n - 1} + R_{n - 1}), with probability 1 - p let {L_{n}, R_{n}} = {X_{n}, R_{n - 1}} or = {L_{n - 1}, X_{n}} according as X_{n} ≥ 1 2(L_{n - 1} + R_{n - 1}) or < 1 2(L_{n - 1} + R_{n - 1}), and let X_{n + 1} be a uniform random variable over {L_{n}, R_{n}}. By this iterated procedure, a random sequence {X_{n}}_{n ≥ 1} is constructed, and it is easy to see that X_{n} converges to a random variable Y_{p} (say) almost surely as n → ∞. Then what is the distribution of Y_{p}? It is shown that the Beta, (2, 2) distribution is the distribution of Y_{1}; that is, the probability density function of Y_{1} is g(y) = 6y(1 - y) I_{0,1}(y). It is also shown that the distribution of Y_{0} is not a known distribution but has some interesting properties (convexity and differentiability).

Original language | English (US) |
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Pages (from-to) | 221-230 |

Number of pages | 10 |

Journal | Journal of Multivariate Analysis |

Volume | 14 |

Issue number | 2 |

DOIs | |

State | Published - Apr 1984 |

### Keywords

- Beta distribution
- characteristic function
- confluent hypergeometric function
- convexity
- differentiability
- moment generating function
- uniform random variable

### ASJC Scopus subject areas

- Statistics and Probability
- Numerical Analysis
- Statistics, Probability and Uncertainty

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## Cite this

*Journal of Multivariate Analysis*,

*14*(2), 221-230. https://doi.org/10.1016/0047-259X(84)90007-1