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. Definition 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 Properties The density f(y) defines a symmetric distribution with median y0, which has tails somewhat heavier than a normal distribution. Special cases In the special case it is the Burr Type XII density. When , which is the density of the standard logistic distribution. Distribution of income 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 x0 = exp(y0) is the median income. If λ = 1, this distribution is often called the Fisk distribution, which has density See also Generalized logistic distribution References Continuous distributions