ON THE MEDIANS OF GAMMA DISTRIBUTIONS AND AN EQUATION OF RAMANUJAN

For n greater-than-or-equal-to 0, let lambda(n) denote the median of the GAMMA(n + 1, 1) distribution. We prove that n + 2/3 < lambda(n) less-than-or-equal-to min(n + log2, n + 2/3 + (2n + 2)-1). These bounds are sharp. There is an intimate relationship between lambda(n) and an equation of Ramanujan. Based on this relationship, we derive the asymptotic expansion of lambda(n) as follows: lambda(n) = n + 2/3 + 8/405n - 64/5103n2 + 2(7).23/3(9).5(2)n3+.... Let median(Z(mu)) denote the median of a Poisson random variable with mean mu, where the median is defined to be the least integer m such that P(Z(mu) less-than-or-equal-to m) greater-than-or-equal-to 1/2. We show that the bounds on lambda(n) imply mu - log2 less-than-or-equal-to median(Z(mu)) < mu + 1/3. This proves a conjecture of Chen and Rubin. These inequalities are sharp.