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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the medians of gamma distributions and an equation of Ramanujan


Author: K. P. Choi
Journal: Proc. Amer. Math. Soc. 121 (1994), 245-251
MSC: Primary 62E15; Secondary 33B15, 41A58
MathSciNet review: 1195477
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Abstract | References | Similar Articles | Additional Information

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:

$\displaystyle \lambda (n) = n + \frac{2}{3} + \frac{8}{{405n}} - \frac{{64}}{{5103{n^2}}} + \frac{{{2^7} \cdot 23}}{{{3^9} \cdot {5^2}{n^3}}} + \cdots .$

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

$\displaystyle \mu - \log 2 \leq {\text{median}}({Z_\mu }) < \mu + \frac{1}{3}.$

This proves a conjecture of Chen and Rubin. These inequalities are sharp.

References [Enhancements On Off] (What's this?)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1195477-8
PII: S 0002-9939(1994)1195477-8
Keywords: Median, Gamma distribution, Poisson distribution, chi-square distribution, Poisson-Gamma relation, Ramanujan's equation
Article copyright: © Copyright 1994 American Mathematical Society