Davis distribution

In statistics, the Davis distributions are a family of continuous probability distributions. It is named after Harold T. Davis (1892–1974), who in 1941 proposed this distribution to model income sizes. (The Theory of Econometrics and Analysis of Economic Time Series). It is a generalization of the Planck's law of radiation from statistical physics.

Davis distribution

Parameters	${\displaystyle b>0}$ scale ${\displaystyle n>0}$ shape ${\displaystyle \mu >0}$ location
Support	${\displaystyle x>\mu }$
PDF	${\displaystyle {\frac {b^{n}{(x-\mu )}^{-1-n}}{\left(e^{\frac {b}{x-\mu }}-1\right)\Gamma (n)\zeta (n)}}}$ Where ${\displaystyle \Gamma (n)}$ is the Gamma function and ${\displaystyle \zeta (n)}$ is the Riemann zeta function
Mean	${\displaystyle {\begin{cases}\mu +{\frac {b\zeta (n-1)}{(n-1)\zeta (n)}}&{\text{if}}\ n>2\\{\text{Indeterminate}}&{\text{otherwise}}\ \end{cases}}}$
Variance	${\displaystyle {\begin{cases}{\frac {b^{2}\left(-(n-2){\zeta (n-1)}^{2}+(n-1)\zeta (n-2)\zeta (n)\right)}{(n-2){(n-1)}^{2}{\zeta (n)}^{2}}}&{\text{if}}\ n>3\\{\text{Indeterminate}}&{\text{otherwise}}\ \end{cases}}}$