The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a power law probability distribution found in a large number of real-world situations (Wikipedia 2006). Outside the field of economics it is generally referred to as the Bradford distribution (Wikipedia 2006).
The Pareto distribution was originally used to describe the allocation of wealth among individuals since it seemed to show rather well the way that a larger portion of the wealth of any society is owned by a smaller percentage of the people in that society (Wikipedia 2006). It can be shown, (Wikipedia 2006), that from a probability density function (PDF) graph of the population f(x), the probability, or fraction, of f(x) that own a small amount of wealth per person, is high. The probability then decreases steadily as wealth increases (Wikipedia 2006).
This distribution is not limited to describing wealth or income distribution, but to many situations in which an equilibrium is found in the distribution of the "small" to the "large" (Wikipedia 2006). The following examples, taken from (Wikipedia 2006), are sometimes seen as approximately Pareto-distributed:
Examples:
- Frequencies of words in longer texts
- The size of human settlements (few cities, many hamlets/villages)
- File size distribution of Internet traffic which uses the TCP protocol (many smaller files, few larger ones)
- Clusters of Bose-Einstein condensate near absolute zero
- The value of oil reserves in oil fields (a few large fields, many small fields)
- The length distribution in jobs assigned supercomputers (a few large ones, many small ones)
- The standardized price returns on individual stocks
- Size of sand particles
- Size of meteorites
- Number of species per genus (please note the subjectivity involved: The tendency to divide a genus into two or more increases with the number of species in it)
- Areas burnt in forest fires
Mathematical properties of the Pareto distribution
If X is a random variable with a Pareto distribution, then the probability that X is greater than some number x is given by:
for all x ≥ xm, where xm is the (necessarily positive) minimum possible value of X, and k is a positive parameter (Wikipedia 2006).
The family of Pareto distributions is parameterized by two quantities, xm and k (Wikipedia 2006). When this distribution is used to model the distribution of wealth, then the parameter k is called the Pareto index (Wikipedia 2006).
It follows that the probability density function is
Pareto distributions are continuous probability distributions (Wikipedia 2006).
The expected value of a random variable following a Pareto distribution is
(if k ≤ 1, the expected value is infinite). Its varianceis
(Note: if
but they are only defined for k > n (Wikipedia 2006). This means that the moment generating function, which is just a Taylor series in x with μn' / n! as coefficients, is not defined (Wikipedia 2006). The characteristic function is given by:
where Γ(a,x) is the incomplete Gamma function (Wikipedia 2006). The Pareto distribution is related to the exponential distribution by:
The Dirac delta function is a limiting case of the Pareto distribution:
References:
Wikipedia. "Pareto distribution", Wikimedia Foundation Inc, http://en.wikipedia.org/wiki/Pareto_distribution, Last accessed 14 February 2006.
Brandon Hodgson
