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Proof of the Chen–Rubin conjecture

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Abstract:

Let n  0 be an integer and let (n) be the median of the Gamma distribution of order n + 1 with parameter 1. In 1986, Chen and Rubin conjectured that n ↦  (n) − n (n = 0, 1, 2, …) is decreasing. We prove the following monotonicity theorem, which settles this conjecture.

Let α and be real numbers. The sequence n ↦  (n) – αn (n = 0, 1, 2, …) is strictly decreasing if and only if α  1. And n ↦ (n) − n (n = 0, 1, 2, …) is strictly increasing if and only if < (1) − log 2 = 0.98519….

Document Type: Research Article

Publication date: August 1, 2005

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