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Probability density function Pareto Type I probability density functions for various α with x_{m} = 1. As α → ∞ the distribution approaches δ(x − x_{m}) where δ is the Dirac delta function.  
Cumulative distribution function Pareto Type I cumulative distribution functions for various α with x_{m} = 1.  
Parameters  x_{m} > 0 scale (real) α > 0 shape (real) 

Support  
CDF  
Mean  
Median  
Mode  
Variance  
Skewness  
Ex. kurtosis  
Entropy  
MGF  
CF  
Fisher information 
Probability density function Pareto Type I probability density functions for various α with x_{m} = 1. As α → ∞ the distribution approaches δ(x − x_{m}) where δ is the Dirac delta function.  
Cumulative distribution function Pareto Type I cumulative distribution functions for various α with x_{m} = 1.  
Parameters  x_{m} > 0 scale (real) α > 0 shape (real) 

Support  
CDF  
Mean  
Median  
Mode  
Variance  
Skewness  
Ex. kurtosis  
Entropy  
MGF  
CF  
Fisher information 
The Pareto distribution, named after the Italian civil engineer and economist Vilfredo Pareto, is a power law probability distribution that is used in description of social, scientific, geophysical, actuarial, and many other types of observable phenomena.
If X is a random variable with a Pareto (Type I) distribution,^{[1]} then the probability that X is greater than some number x, i.e. the survival function (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 x_{m} and a shape parameter 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.
From the definition, the cumulative distribution function of a Pareto random variable with parameters α and x_{m} is
When plotted on linear axes, the distribution assumes the familiar Jshaped curve which approaches each of the orthogonal axes asymptotically. All segments of the curve are selfsimilar (subject to appropriate scaling factors). When plotted in a loglog plot, the distribution is represented by a straight line.
It follows (by differentiation) that the probability density function is
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}.
Suppose X_{1}, X_{2}, X_{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.^{[citation needed]}
The geometric mean (G) is^{[2]}
The harmonic mean (H) is^{[2]}
There is a hierarchy ^{[1]}^{[3]} of Pareto distributions known as Pareto Type I, II, III, IV, and Feller–Pareto distributions.^{[1]}^{[3]}^{[4]} Pareto Type IV contains Pareto Type I–III as special cases. The Feller–Pareto^{[3]}^{[5]} distribution generalizes Pareto Type IV.
The Pareto distribution hierarchy is summarized in the next table comparing the survival functions (complementary CDF).
When μ = 0, the Pareto distribution Type II is also known as the Lomax distribution.^{[6]}
In this section, the symbol x_{m}, used before to indicate the minimum value of x, is replaced by σ.
Support  Parameters  

Type I  
Type II  
Lomax  
Type III  
Type IV 
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 δmoments are finite for some δ > 0, as shown in the table below, where δ is not necessarily an integer.
Condition  Condition  

Type I  
Type II  
Type III  
Type IV 
Feller^{[3]}^{[5]} 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. If
then W has a Feller–Pareto distribution FP(μ, σ, γ, γ_{1}, γ_{2}).^{[1]}
If and are independent Gamma variables, another construction of a Feller–Pareto (FP) variable is^{[7]}
and we write W ~ FP(μ, σ, γ, δ_{1}, δ_{2}). Special cases of the Feller–Pareto distribution are
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. He also used it to describe distribution of income.^{[8]} 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.^{[9]} However, the 8020 rule corresponds to a particular value of α, and in fact, Pareto's data on British income taxes in his Cours d'économie politique indicates that about 30% of the population had about 70% of the income. The probability density function (PDF) graph at the beginning of this article shows 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. (Note that the Pareto distribution is not realistic for wealth for the lower end. In fact, net worth may even be negative.) 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:
The Pareto distribution is related to the exponential distribution as follows. If X is Paretodistributed with minimum x_{m} and index α, then
is exponentially distributed with rate parameter α. Equivalently, if Y is exponentially distributed with rate α, then
is Paretodistributed with minimum x_{m} and index α.
This can be shown using the standard change of variable techniques:
The last expression is the cumulative distribution function of an exponential distribution with rate α.
Note that the Pareto distribution and lognormal distribution are alternative distributions for describing the same types of quantities. One of the connections between the two is that they are both the distributions of the exponential of random variables distributed according to other common distributions, respectively the exponential distribution and normal distribution.^{[citation needed]}
The Pareto distribution is a special case of the generalized Pareto distribution, which is a family of distributions of similar form, but containing an extra parameter in such a way that the support of the distribution is either bounded below (at a variable point), or bounded both above and below (where both are variable), with the Lomax distribution as a special case. This family also contains both the unshifted and shifted exponential distributions.
The Pareto distribution with scale and shape is equivalent to the generalized Pareto distribution with location , scale and shape . Vice versa one can get the Pareto distribution from the GPD by and .
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 "8020 law", according to which 20% of all people receive 80% of all income, and 20% of the most affluent 20% receive 80% of that 80%, and so on, holds precisely when the Pareto index is α = log_{4}(5) = log(5)/log(4), approximately 1.161. This result can be derived from the Lorenz curve formula given below. Moreover, the following have been shown^{[15]} to be mathematically equivalent:
This does not apply only to income, but also to wealth, or to anything else that can be modeled by this distribution.
This excludes Pareto distributions in which 0 < α ≤ 1, which, as noted above, have infinite expected value, and so cannot reasonably model income 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 f 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 x ≥ x_{m}, 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:^{[16]}
Malik (1970)^{[17]} gives the exact joint distribution of . In particular, and are independent and is Pareto with scale parameter x_{m} and shape parameter nα, whereas has an Inversegamma distribution with shape and scale parameters n−1 and nα, respectively.
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: 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.
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 T given by
is Paretodistributed.^{[18]} If U is uniformly distributed on [0, 1), it can be exchanged with (1 − U).
Parameters  location (real) 

Support  
CDF  
Mean  
Median  
Variance  (this is the second moment, NOT the variance)^{[citation needed]} 
Skewness  (this is a formula for the kth moment, NOT the skewness)^{[citation needed]} 
The bounded (or truncated) 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 the second moment).
The probability density function is
where L ≤ x ≤ H, and α > 0.
If U is uniformly distributed on (0, 1), then applying inversetransform method ^{[19]}
is a bounded Paretodistributed.^{[citation needed]}
The symmetric Pareto distribution can be defined by the probability density function:^{[20]}
It has a similar shape to a Pareto distribution for x > x_{m} and is mirror symmetric about the vertical axis.
