Proof of the Chen–Rubin conjecture

Abstract

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

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