Champernowne distribution

The Champernowne distribution has a probability density function given by : f(y;\alpha, \lambda, y_0 ) = \frac{n}{\cosh[\alpha(y - y_0)] + \lambda}, \qquad -\infty where \alpha, \lambda, y_0 are positive parameters, and n is the normalizing constant, which depends on the parameters. The density may be rewritten as : f(y) = \frac{n}{\tfrac 1 2 e^{\alpha(y-y_0)} + \lambda + \tfrac 12 e^{-\alpha(y-y_0)}}, using the fact that \cosh x = \tfrac 1 2 (e^x + e^{-x}). 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 \lambda = 0 (\alpha = \tfrac \pi 2, y_0 = 0) it is the hyperbolic secant distribution. In the special case \lambda=1 it is the Burr Type XII density. When y_0 = 0, \alpha=1, \lambda=1 , : f(y) = \frac{1}{e^y + 2 + e^{-y}} = \frac{e^y}{(1+e^y)^2}, 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 which has density : f(x) = \frac{\alpha x^{\alpha - 1}}{x_0^\alpha [1 + (x/x_0)^\alpha]^2}, \qquad x > 0. ==See also==