Sharp estimates for the median of the Γ(n+1,1) distribution

We give sharp upper and lower bounds for the median of the Γ(n+1,1) distribution, thus providing an immediate proof of two conjectures by Chen and Rubin (Statist. Probab. Lett. 4 (1986) 281) referring to the median of the Poisson distribution. Our approach uses a differential calculus for nonnecessarily smooth functions of the standard Poisson process and the central limit theorem.