Monotonicity of the difference between median and mean of Gamma distributions and of a related Ramanujan sequence

Sven Erick Alm

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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}
2004-12-01, Sven Erick Alm, sea@math.uu.se