The Pareto distribution, named after the Italian
civil engineer,
economist
An economist is a professional and practitioner in the social science discipline of economics.
The individual may also study, develop, and apply theories and concepts from economics and write about economic policy. Within this field there are ...
, and
sociologist Vilfredo Pareto
Vilfredo Federico Damaso Pareto ( , , , ; born Wilfried Fritz Pareto; 15 July 1848 – 19 August 1923) was an Italians, Italian polymath (civil engineer, sociologist, economist, political scientist, and philosopher). He made several important ...
( ), is a
power-law probability distribution that is used in description of
social
Social organisms, including human(s), live collectively in interacting populations. This interaction is considered social whether they are aware of it or not, and whether the exchange is voluntary or not.
Etymology
The word "social" derives from ...
,
quality control,
scientific
Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe.
Science may be as old as the human species, and some of the earliest archeological evidence for ...
,
geophysical
Geophysics () is a subject of natural science concerned with the physical processes and physical properties of the Earth and its surrounding space environment, and the use of quantitative methods for their analysis. The term ''geophysics'' some ...
,
actuarial, and many other types of observable phenomena; the principle originally applied to describing the
distribution of wealth
The distribution of wealth is a comparison of the wealth of various members or groups in a society. It shows one aspect of economic inequality or economic heterogeneity.
The distribution of wealth differs from the income distribution in that ...
in a society, fitting the trend that a large portion of wealth is held by a small fraction of the population.
The
Pareto principle or "80-20 rule" stating that 80% of outcomes are due to 20% of causes was named in honour of Pareto, but the concepts are distinct, and only Pareto distributions with shape value () of log
45 ≈ 1.16 precisely reflect it. Empirical observation has shown that this 80-20 distribution fits a wide range of cases, including natural phenomena and human activities.
Definitions
If ''X'' is a
random variable with a Pareto (Type I) distribution,
then the probability that ''X'' is greater than some number ''x'', i.e. the
survival function
The survival function is a function that gives the probability that a patient, device, or other object of interest will survive past a certain time.
The survival function is also known as the survivor function
or reliability function.
The te ...
(also called tail function), is given by
:
where ''x''
m is the (necessarily positive) minimum possible value of ''X'', and ''α'' is a positive parameter. The Pareto Type I distribution is characterized by a
scale parameter
In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions. The larger the scale parameter, the more spread out the distribution.
Definition
If a family o ...
''x''
m and a
shape parameter
In probability theory and statistics, a shape parameter (also known as form parameter) is a kind of numerical parameter of a parametric family of probability distributionsEveritt B.S. (2002) Cambridge Dictionary of Statistics. 2nd Edition. CUP.
t ...
''α'', which is known as the ''tail index''. When this distribution is used to model the distribution of wealth, then the parameter ''α'' is called the
Pareto index
Pareto may refer to:
People
* Vilfredo Pareto (1848–1923), Italian economist, political scientist, and philosopher, works named for him include:
** Pareto analysis, a statistical analysis tool in problem solving
**Pareto distribution, a power-l ...
.
Cumulative distribution function
From the definition, the
cumulative distribution function of a Pareto random variable with parameters ''α'' and ''x''
m is
:
Probability density function
It follows (by
differentiation) that the
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 ...
is
:
When plotted on linear axes, the distribution assumes the familiar J-shaped curve which approaches each of the orthogonal axes
asymptotically. All segments of the curve are self-similar (subject to appropriate scaling factors). When plotted in a
log-log plot, the distribution is represented by a straight line.
Properties
Moments and characteristic function
* The
expected value of a
random variable following a Pareto distribution is
:
::
* The
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
of a
random variable following a Pareto distribution is
::
: (If ''α'' ≤ 1, the variance does not exist.)
* The raw
moment (mathematics), moments are
::
* The
moment generating function
In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. Thus, it provides the basis of an alternative route to analytical results compare ...
is only defined for non-positive values ''t'' ≤ 0 as
::
::
Thus, since the expectation does not converge on an
open interval containing
we say that the moment generating function does not exist.
* The
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
* The indicator function of a subset, that is the function
::\mathbf_A\colon X \to \,
:which for a given subset ''A'' of ''X'', has value 1 at points ...
is given by
::
: where Γ(''a'', ''x'') is the
incomplete gamma function.
The parameters may be solved for using the
method of moments.
Conditional distributions
The
conditional probability distribution
In probability theory and statistics, given two jointly distributed random variables X and Y, the conditional probability distribution of Y given X is the probability distribution of Y when X is known to be a particular value; in some cases the ...
of a Pareto-distributed random variable, given the event that it is greater than or equal to a particular number
exceeding
, is a Pareto distribution with the same Pareto index
but with minimum
instead of
. This implies that the conditional expected value (if it is finite, i.e.
) is proportional to
. In case of random variables that describe the lifetime of an object, this means that life expectancy is proportional to age, and is called the
Lindy effect
The Lindy effect (also known as Lindy's Law) is a theorized phenomenon by which the future life expectancy of some non-perishable things, like a technology or an idea, is proportional to their current age. Thus, the Lindy effect proposes the longe ...
or Lindy's Law.
A characterization theorem
Suppose
are
independent identically distributed
In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is us ...
random variables whose probability distribution is supported on the interval
for some
. Suppose that for all
, the two random variables
and
are independent. Then the common distribution is a Pareto distribution.
Geometric mean
The geometric mean (''G'') is
[Johnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions Vol 1. Wiley Series in Probability and Statistics.]
:
Harmonic mean
The
harmonic mean (''H'') is
:
Graphical representation
The characteristic curved '
long tail' distribution when plotted on a linear scale, masks the underlying simplicity of the function when plotted on a
log-log graph, which then takes the form of a straight line with negative gradient: It follows from the formula for the probability density function that for ''x'' ≥ ''x''
m,
:
Since ''α'' is positive, the gradient −(''α'' + 1) is negative.
Related distributions
Generalized Pareto distributions
There is a hierarchy
[Johnson, Kotz, and Balakrishnan (1994), (20.4).] of Pareto distributions known as Pareto Type I, II, III, IV, and Feller–Pareto distributions. Pareto Type IV contains Pareto Type I–III as special cases. The Feller–Pareto[ "The densities (4.3) are sometimes called after the economist ''Pareto''. It was thought (rather naïvely from a modern statistical standpoint) that income distributions should have a tail with a density ~ ''Ax''−''α'' as ''x'' → ∞."] distribution generalizes Pareto Type IV.
Pareto types I–IV
The Pareto distribution hierarchy is summarized in the next table comparing the survival function
The survival function is a function that gives the probability that a patient, device, or other object of interest will survive past a certain time.
The survival function is also known as the survivor function
or reliability function.
The te ...
s (complementary CDF).
When ''μ'' = 0, the Pareto distribution Type II is also known as the Lomax distribution
The Lomax distribution, conditionally also called the Pareto Type II distribution, is a heavy-tail probability distribution used in business, economics, actuarial science, queueing theory and Internet traffic modeling. It is named after K.  ...
.
In this section, the symbol ''x''m, used before to indicate the minimum value of ''x'', is replaced by ''σ''.
The shape parameter ''α'' is the tail index, ''μ'' is location, ''σ'' is scale, ''γ'' is an inequality parameter. Some special cases of Pareto Type (IV) are
::
::
::
The finiteness of the mean, and the existence and the finiteness of the variance depend on the tail index ''α'' (inequality index ''γ''). In particular, fractional ''δ''-moments are finite for some ''δ'' > 0, as shown in the table below, where ''δ'' is not necessarily an integer.
Feller–Pareto distribution
Feller[ defines a Pareto variable by transformation ''U'' = ''Y''−1 − 1 of a beta random variable ''Y'', whose probability density function is
:
where ''B''( ) is the ]beta function
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral
: \Beta(z_1,z_2) = \int_0^1 t^( ...
. If
:
then ''W'' has a Feller–Pareto distribution FP(''μ'', ''σ'', ''γ'', ''γ''1, ''γ''2).[
If and are independent Gamma variables, another construction of a Feller–Pareto (FP) variable is
:
and we write ''W'' ~ FP(''μ'', ''σ'', ''γ'', ''δ''1, ''δ''2). Special cases of the Feller–Pareto distribution are
:
:
:
:
]
Inverse-Pareto Distribution / Power Distribution
When a random variable follows a pareto distribution, then its inverse follows an Inverse Pareto distribution.
Inverse Pareto distribution is equivalent to a Power distribution
Electric power distribution is the final stage in the delivery of electric power; it carries electricity from the transmission system to individual consumers. Distribution substations connect to the transmission system and lower the transmissi ...
:
Relation to the exponential distribution
The Pareto distribution is related to the exponential distribution as follows. If ''X'' is Pareto-distributed with minimum ''x''m and index ''α'', then
:
is exponentially distributed
In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant averag ...
with rate parameter ''α''. Equivalently, if ''Y'' is exponentially distributed with rate ''α'', then
:
is Pareto-distributed with minimum ''x''m and index ''α''.
This can be shown using the standard change-of-variable techniques:
: