In statistics, the Champernowne distribution is a symmetric, continuous probability distribution, describing random variables that take both positive and negative values. It is a generalization of the logistic distribution that was introduced by D. G. Champernowne. Champernowne developed the distribution to describe the logarithm of income.
The Champernowne distribution has a probability density function given by
where are positive parameters, and n is the normalizing constant, which depends on the parameters. The density may be rewritten as
using the fact that
The density f(y) defines a symmetric distribution with median y<sub>0</sub>, which has tails somewhat heavier than a normal distribution.
In the special case () it is the hyperbolic secant distribution.
In the special case it is the Burr Type XII density.
When ,
which is the density of the standard logistic distribution.
If the distribution of Y, the logarithm of income, has a Champernowne distribution, then the density function of the income X = exp(Y) is
where x<sub>0</sub> = exp(y<sub>0</sub>) is the median income. If û = 1, this distribution is often called the Fisk distribution, which has density