# Connexions

You are here: Home » Content » What is Pareto distribution?

### Recently Viewed

This feature requires Javascript to be enabled.

# What is Pareto distribution?

Module by: Brandon Hodgson. E-mail the author

## What is the Pareto distribution?

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:

(1)(Wikipedia 2006)

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

(2)(Wikipedia 2006)

Pareto distributions are continuous probability distributions (Wikipedia 2006).

The expected value of a random variable following a Pareto distribution is

(3)(Wikipedia 2006)

(if k ≤ 1, the expected value is infinite). Its varianceis

(4)(Wikipedia 2006)

(Note: if

, the variance is infinite) (Wikipedia 2006). The raw moments are found to be:

(5)(Wikipedia 2006)

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:

(6)(Wikipedia 2006)

where Γ(a,x) is the incomplete Gamma function (Wikipedia 2006). The Pareto distribution is related to the exponential distribution by:

(7)(Wikipedia 2006)

The Dirac delta function is a limiting case of the Pareto distribution:

(8)(Wikipedia 2006)

References:

Wikipedia. "Pareto distribution", Wikimedia Foundation Inc, http://en.wikipedia.org/wiki/Pareto_distribution, Last accessed 14 February 2006.

Brandon Hodgson

## Content actions

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks