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Monotonicity of the difference between median and mean of gamma distributions and of a related Ramanujan sequencePrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2003 (English)In: Bernoulli, ISSN 1350-7265, Vol. 9, no 2, 351-371 p.Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

2003. Vol. 9, no 2, 351-371 p.
##### Keyword [en]

Gamma distribution, mean, median, Poisson distribution, Ramanujan
##### National Category

Probability Theory and Statistics
##### Identifiers

URN: urn:nbn:se:uu:diva-22346OAI: oai:DiVA.org:uu-22346DiVA: diva2:50119
#####

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Available from: 2007-01-16 Created: 2007-01-16 Last updated: 2011-01-13

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 $\frac2{21}<k_n\le\frac8{45}$. We also show that the sequence $\{k_n\}$ is decreasing.

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