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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the medians of gamma distributions and an equation of Ramanujan
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by K. P. Choi PDF
Proc. Amer. Math. Soc. 121 (1994), 245-251 Request permission

Abstract:

For $n \geq 0$, let $\lambda (n)$ denote the median of the $\Gamma (n + 1,1)$ distribution. We prove that $n + \tfrac {2}{3} < \lambda (n) \leq \min (n + \log 2, n + \tfrac {2}{3} + {(2n + 2)^{ - 1}})$. These bounds are sharp. There is an intimate relationship between $\lambda (n)$ and an equation of Ramanujan. Based on this relationship, we derive the asymptotic expansion of $\lambda (n)$ as follows: \[ \lambda (n) = n + \frac {2}{3} + \frac {8}{{405n}} - \frac {{64}}{{5103{n^2}}} + \frac {{{2^7} \cdot 23}}{{{3^9} \cdot {5^2}{n^3}}} + \cdots .\] Let median $({Z_\mu })$ denote the median of a Poisson random variable with mean $\mu$, where the median is defined to be the least integer m such that $P({Z_\mu } \leq m) \geq \tfrac {1}{2}$. We show that the bounds on $\lambda (n)$ imply \[ \mu - \log 2 \leq {\text {median}}({Z_\mu }) < \mu + \frac {1}{3}.\] This proves a conjecture of Chen and Rubin. These inequalities are sharp.
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Additional Information
  • © Copyright 1994 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 121 (1994), 245-251
  • MSC: Primary 62E15; Secondary 33B15, 41A58
  • DOI: https://doi.org/10.1090/S0002-9939-1994-1195477-8
  • MathSciNet review: 1195477