13rd  Top probability distributions 
12nd  Top basic probability topics 
Probability density function Pareto probability density functions for various α with x_{m} = 1. The horizontal axis is the x parameter. As α → ∞ the distribution approaches δ(x − x_{m}) where δ is the Dirac delta function. 

Cumulative distribution function Pareto cumulative distribution functions for various α with x_{m} = 1. The horizontal axis is the x parameter. 

parameters: 
scale (real) shape (real) 

support:  
pdf:  
cdf:  
mean:  
median:  
mode:  
variance:  
skewness:  
kurtosis:  
entropy:  
mgf:  
cf: 
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 "8020 rule" which says that 20% of the population controls 80% of the wealth^{[1]}. It can be seen from the probability density function (PDF) graph on the right, that the "probability" or fraction of the population that owns a small amount of wealth per person is rather high, and then decreases steadily as wealth increases. This distribution is not limited to describing wealth or income, 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 Paretodistributed:
If X is a random variable with a Pareto distribution, then the probability that X is greater than some number x is given by
where x_{m} is the (necessarily positive) minimum possible value of X, and α is a positive parameter. The family of Pareto distributions is parameterized by two quantities, x_{m} and α. When this distribution is used to model the distribution of wealth, then the parameter α is called the Pareto index.
It follows from the above that therefore the cumulative distribution function of a Pareto random variable with parameters α and x_{m} is
It follows (by differentiation) that the probability density function is
The Dirac delta function is a limiting case of the Pareto density:
The conditional probability distribution of a Paretodistributed random variable, given the event that it is greater than or equal to a particular number x_{1} exceeding x_{m}, is a Pareto distribution with the same Pareto index α but with minimum x_{1} instead of x_{m}.
The Pareto distribution is related to the exponential distribution as follows. Suppose X is Paretodistributed with minimum x_{m} and index α. Let
Then Y is exponentially distributed with intensity α, or equivalently with expected value 1/α:
Equivalently, if Y is exponentially distributed with intensity α, then
is Paretodistributed with minimum x_{m} and index α.
Suppose X_{i}, i = 1, 2, 3, ... are independent identically distributed random variables whose probability distribution is supported on the interval [x_{m}, ∞) for some x_{m} > 0. Suppose that for all n, the two random variables min{ X_{1}, ..., X_{ n} } and (X_{1} + ... + X_{ n})/min{ X_{1}, ..., X_{ n} } are independent. Then the common distribution is a Pareto distribution.
Pareto distributions are continuous probability distributions. Zipf's law, also sometimes called the zeta distribution, may be thought of as a discrete counterpart of the Pareto distribution.
The Lorenz curve is often used to characterize income and wealth distributions. For any distribution, the Lorenz curve L(F) is written in terms of the PDF ƒ or the CDF F as
where x(F) is the inverse of the CDF. For the Pareto distribution,
and the Lorenz curve is calculated to be
where α must be greater than or equal to unity, since the denominator in the expression for L(F) is just the mean value of x. Examples of the Lorenz curve for a number of Pareto distributions are shown in the graph on the right.
The Gini coefficient is a measure of the deviation of the Lorenz curve from the equidistribution line which is a line connecting [0, 0] and [1, 1], which is shown in black (α = ∞) in the Lorenz plot on the right. Specifically, the Gini coefficient is twice the area between the Lorenz curve and the equidistribution line. The Gini coefficient for the Pareto distribution is then calculated to be
(see Aaberge 2005).
The likelihood function for the Pareto distribution parameters α and x_{m}, given a sample x = (x_{1}, x_{2}, ..., x_{n}), is
Therefore, the logarithmic likelihood function is
It can be seen that is monotonically increasing with x_{m}, that is, the greater the value of x_{m}, the greater the value of the likelihood function. Hence, since , we conclude that
To find the estimator for α, we compute the corresponding partial derivative and determine where it is zero:
Thus the maximum likelihood estimator for α is:
The expected statistical error is:
The characteristic curved 'long tail' distribution when plotted on a linear scale, masks the underlying simplicity of the function when plotted on a loglog graph, which then takes the form of a straight line with negative gradient.
Random samples can be generated using inverse transform sampling. Given a random variate U drawn from the uniform distribution on the unit interval (0, 1), the variate
is Paretodistributed.[2]
The bounded Pareto distribution has three parameters α, L and H. As in the standard Pareto distribution α determines the shape. L denotes the minimal value, and H denotes the maximal value. (The Variance in the table on the right should be interpreted as 2nd Moment).
parameters: 
location (real) 

support:  
pdf:  
cdf:  
mean:  
median:  
mode:  
variance:  
skewness:  
kurtosis:  
entropy:  
mgf:  
cf: 
The probability density function is
where , and α > 0.
If U is uniformly distributed on (0, 1), then
is bounded Paretodistributed [3]
The family of generalized Pareto distributions (GPD) has three parameters and .
parameters: 
location (real) 

support:  
pdf:  where 
cdf:  
mean:  
median:  
mode:  
variance:  
skewness:  
kurtosis:  
entropy:  
mgf:  
cf: 
The cumulative distribution function is
for , and when , where is the location parameter, the scale parameter and the shape parameter. Note that some references give the "shape parameter" as .
The probability density function is:
or
again, for ,
and
when
.
If U is uniformly distributed on (0, 1], then

Singular 
Plural 
Pareto distribution (plural Pareto distributions)

