Pareto Distribution

The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a power law probability distribution that coincides with social, scientific, geophysical, actuarial, and many other types of observable phenomena. Outside the field of economics it is at times referred to as the Bradford distribution.

Pareto originally used this distribution 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. This idea is sometimes expressed more simply as the Pareto principle or the “80-20 rule” which says that 20% of the population owns 80% of the wealth. It can be seen from the probability density function (PDF) graph on the right, that the “probability” or fraction of the population f(x) that owns a small amount of wealth per person (x) is rather high, and then decreases steadily as wealth increases. 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”. The following examples are sometimes seen as approximately Pareto-distributed:

• Frequencies of words in longer texts (a few words are used often, lots of words are used infrequently)
• The sizes 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 values 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
• Sizes of sand particles
• Sizes of meteorites
• Numbers of species per genus (There is 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
• Severity of large casualty losses for certain lines of business such as general liability, commercial auto, and workers compensation.

The probability distribution function of this distribution is

f(x,a,s) = sa^s/x^(s+1)

support a ⇐ x ⇐ inf