On the medians of gamma distributions and an equation of Ramanujan
Author:
K. P. Choi
Journal:
Proc. Amer. Math. Soc. 121 (1994), 245251
MSC:
Primary 62E15; Secondary 33B15, 41A58
MathSciNet review:
1195477
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Abstract 
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Abstract: For , let denote the median of the distribution. We prove that . These bounds are sharp. There is an intimate relationship between and an equation of Ramanujan. Based on this relationship, we derive the asymptotic expansion of as follows: Let median denote the median of a Poisson random variable with mean , where the median is defined to be the least integer m such that . We show that the bounds on imply This proves a conjecture of Chen and Rubin. These inequalities are sharp.
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 H. Dinges, Special cases of second order Wiener Germ approximations, Probab. Theory Related Fields 83 (1989), 557. MR 1012493 (91h:60026)
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 A. T. Doodson, Relation of the mode, median and mean in frequency curves, Biometrika 11 (1971), 425429.
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 J. C. W. Marsaglia, The incomplete Gamma function and Ramanujan's rational approximation to , J. Statist. Comput. Simulation 24 (1986), 163168.
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 S. Ramanujan, J. Indian Math. Soc. 3 (1911), 151152.
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 , Collected Papers, Chelsea, New York, 1927.
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 G. Szegö, Über einige von S. Ramanujan gestelle Aufgaben, J. London Math. Soc. 3 (1928), 225232.
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 G. N. Watson, Theorems stated by Ramanujan (V): Approximations connected with , Proc. London Math. Soc. (2) 29 (1927), 293308.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199411954778
PII:
S 00029939(1994)11954778
Keywords:
Median,
Gamma distribution,
Poisson distribution,
chisquare distribution,
PoissonGamma relation,
Ramanujan's equation
Article copyright:
© Copyright 1994
American Mathematical Society
