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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
DOI: https://doi.org/10.1090/S0002-9939-1994-1195477-8
MathSciNet review: 1195477
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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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Keywords: Median, Gamma distribution, Poisson distribution, chi-square distribution, Poisson-Gamma relation, Ramanujan’s equation
Article copyright: © Copyright 1994 American Mathematical Society