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On the Medians of Gamma Distributions and an Equation of Ramanujan

K. P. Choi
Proceedings of the American Mathematical Society
Vol. 121, No. 1 (May, 1994), pp. 245-251
DOI: 10.2307/2160389
Stable URL: http://www.jstor.org/stable/2160389
Page Count: 7
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On the Medians of Gamma Distributions and an Equation of Ramanujan
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Abstract

For n ≥ 0, let λ(n) denote the median of the Γ(n + 1, 1) distribution. We prove that $n + \frac{2}{3} < \lambda(n) \leq \min(n + \log 2, n + \frac{2}{3} + (2n + 2)^{-1})$. These bounds are sharp. There is an intimate relationship between λ(n) and an equation of Ramanujan. Based on this relationship, we derive the asymptotic expansion of λ(n) as follows: λ(n) = n + 2/3 + 8/405n - 64/5103n2 + 27 · 23/39 · 52n 3 + .... Let median(Zμ) denote the median of a Poisson random variable with mean μ, where the median is defined to be the least integer m such that P(Zμ ≤ m) ≥ 1/2. We show that the bounds on λ(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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