Champernowne distribution

The Champernowne distribution has a probability density function given by

${\displaystyle f(y;\alpha ,\lambda ,y_{0})={\frac {n}{\cosh[\alpha (y-y_{0})]+\lambda }},\qquad -\infty <y<\infty ,}$

where ${\displaystyle \alpha ,\lambda ,y_{0}}$ are positive parameters, and n is the normalizing constant, which depends on the parameters. The density may be rewritten as

${\displaystyle f(y)={\frac {n}{1/2e^{\alpha (y-y_{0})}+\lambda +1/2e^{-\alpha (y-y_{0})}}},}$

using the fact that ${\displaystyle \cosh y=(e^{y}+e^{-y})/2.}$

If the distribution of Y, the logarithm of income, has a Champernowne distribution, then the density function of the income X = exp(Y) is[1]

${\displaystyle f(x)={\frac {n}{x[1/2(x/x_{0})^{-\alpha }+\lambda +a/2(x/x_{0})^{\alpha }]}},\qquad x>0,}$

where x₀ = exp(y₀) is the median income. If λ = 1, this distribution is often called the Fisk distribution,[4] which has density

${\displaystyle f(x)={\frac {\alpha x^{\alpha -1}}{x_{0}^{\alpha }[1+(x/x_{0})^{\alpha }]^{2}}},\qquad x>0.}$

• Generalized logistic distribution