Monotonicity of the difference between median and mean of Gamma distributions and of a related Ramanujan sequence
by
Sven Erick Alm
Uppsala University
U.U.D.M. Report 2002:5 ISSN 1101-3591
In Bernoulli 9(2), 2003, 351-371.
pdf
Abstract
For $n\ge0$, let $\lambda_n$ be the median
of the $\Gamma(n+1,1)$ distribution. We prove that the sequence
$\{\alpha_n=\lambda_n-n\}$ decreases from $\log 2$ to $2/3$ as $n$ increases from
0 to $\infty$. The difference, $1-\alpha_n$, between the mean and the
median thus increases from $1-\log 2$ to $1/3$.
This result also proves the following conjecture by Chen \& Rubin about the
Poisson distributions: Let $Y_{\mu}\sim\text{Poisson}(\mu)$, and
$\lambda_n$ be the largest $\mu$ such that $P(Y_{\mu}\le n)=1/2$,
then $\lambda_n-n$ is decreasing in $n$.
The sequence $\{\alpha_n\}$ is related to a sequence $\{\theta_n\}$,
introduced by Ramanujan, which is known to be decreasing and of the
form
$\theta_n=\frac13+\frac4{135(n+k_n)}$, where $\frac2{21}