Tight bounds for the median of a gamma distribution

The median of a standard gamma distribution, as a function of its shape parameter k, has no known representation in terms of elementary functions. In this work we prove the tightest upper and lower bounds of the form 2(−1/k)(A + k): an upper bound with A = e(−γ) (with γ being the Euler–Mascheroni constant) and a lower bound with [Image: see text] . These bounds are valid over the entire domain of k > 0, staying between 48 and 55 percentile. We derive and prove several other new tight bounds in support of the proofs.