Davis distribution

The probability density function of the Davis distribution is given by

${\displaystyle f(x;\mu ,b,n)={\frac {b^{n}{(x-\mu )}^{-1-n}}{\left(e^{\frac {b}{x-\mu }}-1\right)\Gamma (n)\zeta (n)}}}$

where ${\displaystyle \Gamma (n)}$ is the Gamma function and ${\displaystyle \zeta (n)}$ is the Riemann zeta function. Here μ, b, and n are parameters of the distribution, and n need not be an integer.

In an attempt to derive an expression that would represent not merely the upper tail of the distribution of income, Davis required an appropriate model with the following properties^[1]

• ${\displaystyle f(\mu )=0\,}$ for some ${\displaystyle \mu >0\,}$
• A modal income exists
• For large x, the density behaves like a Pareto distribution:

${\displaystyle f(x)\sim A{(x-\mu )}^{-\alpha -1}\,.}$

• If ${\displaystyle X\sim \mathrm {Davis} (b=1,n=4,\mu =0)\,}$ then
  ${\displaystyle {\tfrac {1}{X}}\sim \mathrm {Planck} }$ (Planck's law)