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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
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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  • K. O. Bowman and L. R. Shenton, Binomial and Poisson mixtures, maximum likelihood, and Maple code, Far East J. Theor. Stat. 20 (2006), no. 1, 73–95. MR 2279464
  • Jeesen Chen and Herman Rubin, Bounds for the difference between median and mean of gamma and Poisson distributions, Statist. Probab. Lett. 4 (1986), no. 6, 281–283. MR 858317, DOI
  • H. Dinges, Special cases of second order Wiener germ approximations, Probab. Theory Related Fields 83 (1989), no. 1-2, 5–57. MR 1012493, DOI
  • A. T. Doodson, Relation of the mode, median and mean in frequency curves, Biometrika 11 (1971), 425-429.
  • Donald E. Knuth, The art of computer programming, 2nd ed., Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1975. Volume 1: Fundamental algorithms; Addison-Wesley Series in Computer Science and Information Processing. MR 0378456
  • J. C. W. Marsaglia, The incomplete Gamma function and Ramanujan’s rational approximation to ${e^x}$, J. Statist. Comput. Simulation 24 (1986), 163-168. S. Ramanujan, J. Indian Math. Soc. 3 (1911), 151-152. ---, Collected Papers, Chelsea, New York, 1927. G. Szegö, Über einige von S. Ramanujan gestelle Aufgaben, J. London Math. Soc. 3 (1928), 225-232. G. N. Watson, Theorems stated by Ramanujan (V): Approximations connected with ${e^x}$, Proc. London Math. Soc. (2) 29 (1927), 293-308.

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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