Bernoulli

  • Bernoulli
  • Volume 9, Number 2 (2003), 351-371.

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 $n\ge 0$, let $\lambda_n$ be the median of the $\Gamma(n+1,1)$ distribution. We prove that the sequence $\{\alphapha_n=\lambda_n-n\}$ decreases from $\log 2$ to $\frac{2}{3}$ as $n$ increases from 0 to $\infty$. The difference, $1-\alphapha_n$, between the mean and the median thus increases from $1-\log 2$ to $\frac{1}{3}$. This result also proves a conjecture by Chen and Rubin about the Poisson distributions: if $Y_{\mu}\sim\text{Poisson}(\mu)$, and $\lambda_n$ is the largest $\mu$ such that $P(Y_{\mu}\le n)=\frac{1}{2}$, then $\lambda_n-n$ is decreasing in $n$. The sequence $\{\alphapha_n\}$ is related to a sequence $\{\theta_n\}$, introduced by Ramanujan, which is known to be decreasing and of the form $\theta_n=\frac{1}{3}+4/(135(n+k_n))$, where $\frac{2}{21}<k_n\le\frac{8}{45}$. We also show that the sequence $\{k_n\}$ is decreasing.

Article information

Source
Bernoulli Volume 9, Number 2 (2003), 351-371.

Dates
First available in Project Euclid: 6 November 2003

Permanent link to this document
http://projecteuclid.org/euclid.bj/1068128981

Digital Object Identifier
doi:10.3150/bj/1068128981

Mathematical Reviews number (MathSciNet)
MR1997033

Zentralblatt MATH identifier
1015.62007

Keywords
gamma distribution mean median Poisson distribution Ramanujan

Citation

Erick Alm, Sven. Monotonicity of the difference between median and mean of gamma distributions and of a related Ramanujan sequence. Bernoulli 9 (2003), no. 2, 351--371. doi:10.3150/bj/1068128981. http://projecteuclid.org/euclid.bj/1068128981.


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