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The density $$f(s)\propto \frac{s}{s+\alpha}e^{-s},\quad s > 0$$ where $\alpha \ge 0$ is a parameter, lives between the exponential ($\alpha=0$) and $\Gamma(2,1)$ ($\alpha \to \infty$) distributions. Just curious if this happens to be an example of a more general family of distributions? I do not recognize it as such.

kjetil b halvorsen
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nth
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1 Answers1

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The density function becomes $$ f(s) = {\frac {\alpha}{1-2\,{{\rm e}^{\alpha}}{\it Ei} \left( 3,\alpha \right) }}\cdot \frac{s}{s+\alpha} e^{-s}, \quad s>0 $$ where ${\it Ei}$ is the exponential integral.

I cannot recognize that as something having a known name. Where did you encounter this?

kjetil b halvorsen
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    I needed to define it in the course of working on problem in mathematical genetics. It would not surprise me to find that it is relatively unknown -- always good to double-check though! – nth Jul 21 '19 at 16:53
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    This math search engine http://www.searchonmath.com/result?query=%24+%5Cfrac%7Bs%7D%7Bs%2B%5Calpha%7De%5E%7B-s%7D%24&page=1&tm=0&domains= do not find anything. – kjetil b halvorsen Jul 21 '19 at 17:09
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    +1. Do you have any guidance as to how to profitably use this engine? I just tried https://www.searchonmath.com/result?query=Gauss+%24%5Cfrac%7B1%7D%7B%5Csqrt%7B2%5Cpi%7D%7De%5E%7B-%5Cfrac%7B1%7D%7B2%7Dx%5E2%7D%24&page=1&tm=0&domains=, which, imho, should return results... – Christoph Hanck Nov 24 '20 at 11:13
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    @Cristoph Hanck: 1) seems like it requires registration now? 2) the syntax seems to be very strict. Following their first example `$E=mc^2$` gives no results, but `${E=mc^2}$` works. – kjetil b halvorsen Nov 24 '20 at 11:33