In probability

Probability

Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

 and statistics

Statistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, the parabolic fractal distribution is a type of discrete probability distribution in which the logarithm of the frequency or size of entities in a population is a quadratic polynomial

Quadratic polynomial

In mathematics, a quadratic polynomial or quadratic is a polynomial of degree two, also called second-order polynomial. That means the exponents of the polynomial's variables are no larger than 2...

 of the logarithm of the rank. This can markedly improve the fit over a simple power-law relationship (see references below).

In a number of applications, there is a so-called King effect

King effect

In mathematical statistics, economics, and econophysics, the King effect refers to the phenomenon where the top one or two members of a ranked set show up as outliers...

 where the highest-ranked item has a significantly greater frequency or size than the model predicts on the basis of the other items.

Definition

The probability mass function

Probability mass function

In probability theory and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value...

is given, as a function of the rank n, by
where b and c are paramerers of the distribution.