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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles 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.
References
  • 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 10.1016/0167-7152(86)90044-1
  • H. Dinges, Special cases of second order Wiener germ approximations, Probab. Theory Related Fields 83 (1989), no. 1-2, 5–57. MR 1012493, DOI 10.1007/BF00333142
    • 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 Series in Computer Science and Information Processing, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1975. Volume 1: Fundamental algorithms. 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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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