In
statistics, the Champernowne distribution is a symmetric,
continuous probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
, describing
random variables that take both positive and negative values. It is a generalization of the
logistic distribution
Logistic may refer to:
Mathematics
* Logistic function, a sigmoid function used in many fields
** Logistic map, a recurrence relation that sometimes exhibits chaos
** Logistic regression, a statistical model using the logistic function
** Logit, ...
that was introduced by
D. G. Champernowne.
[ Section 7.3 "Champernowne Distribution."]
/ref> Champernowne developed the distribution to describe the logarithm of income.[
]
Definition
The Champernowne distribution has a probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
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 ''y''0, 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
Logistic may refer to:
Mathematics
* Logistic function, a sigmoid function used in many fields
** Logistic map, a recurrence relation that sometimes exhibits chaos
** Logistic regression, a statistical model using the logistic function
** Logit, ...
.
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 ''x''0 = exp(''y''0) is the median income. If λ = 1, this distribution is often called the Fisk distribution,] which has density
:
See also
* Generalized logistic distribution
References
{{DEFAULTSORT:Champernowne distribution
Continuous distributions