Davis distribution
Parameters	${\displaystyle b>0}$scale ${\displaystyle n>0}$shape ${\displaystyle \mu >0}$location
Support	${\displaystyle x>\mu }$
PDF	${\displaystyle \frac{b^n{(x-\mu )}^{-1-n}}{\left(e^{\frac{b}{x-\mu }}-1\right)\mathrm{\Gamma }(n)\zeta (n)}}$ Where${\displaystyle \mathrm{\Gamma }(n)}$ is theGamma function and${\displaystyle \zeta (n)}$ is theRiemann zeta function
Mean
Variance

Instatistics, theDavis distributions are a family ofcontinuous probability distributions. It is named afterHarold 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 thePlanck's law of radiation fromstatistical physics.

Theprobability density function of the Davis distribution is given by

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

where${\displaystyle \mathrm{\Gamma }(n)}$ is theGamma function and${\displaystyle \zeta (n)}$ is theRiemann zeta function. Here μ,b, andn are parameters of the distribution, andn 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 largex, the density behaves like aPareto distribution:

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

• If${\displaystyle X\sim \mathrm{D}\mathrm{a}\mathrm{v}\mathrm{i}\mathrm{s}(b=1,n=4,\mu =0)}$ then
  ${\displaystyle \frac{1}{X}\sim \mathrm{P}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{k}}$ (Planck's law)

• Kleiber, Christian (2003).Statistical Size Distributions in Economics and Actuarial Sciences. Wiley Series in Probability and Statistics.ISBN 978-0-471-15064-0.
• Davis, H. T. (1941).The Analysis of Economic Time Series. The Principia Press, Bloomington, IndianaDownload book
• Victoria-Feser, Maria-Pia. (1993)Robust methods for personal income distribution models. Thèse de doctorat : Univ. Genève, 1993, no. SES 384 (p. 178)

• Continuous distributions

• Articles with short description
• Short description matches Wikidata