Pareto distribution

Pareto Type I

Probability density function Pareto Type I probability density functions for various α with xm = 1. As α → ∞ the distribution approaches δ(x − xm) where δ is the Dirac delta function.
Cumulative distribution function Pareto Type I cumulative distribution functions for various α with xm = 1.
Parameters	xm > 0 scale (real) α > 0 shape (real)
Support	${\displaystyle x\in [x_{\mathrm {m} },+\infty )}$
PDF	${\displaystyle {\frac {\alpha \,x_{\mathrm {m} }^{\alpha }}{x^{\alpha +1}}}{\text{ for }}x\geq x_{m}}$
CDF	${\displaystyle 1-\left({\frac {x_{\mathrm {m} }}{x}}\right)^{\alpha }{\text{ for }}x\geq x_{m}}$
Mean	${\displaystyle {\begin{cases}\infty &{\text{for }}\alpha \leq 1\\{\frac {\alpha \,x_{\mathrm {m} }}{\alpha -1}}&{\text{for }}\alpha >1\end{cases}}}$
Median	${\displaystyle x_{\mathrm {m} }{\sqrt[{\alpha }]{2}}}$
Mode	${\displaystyle x_{\mathrm {m} }}$
Variance	${\displaystyle {\begin{cases}\infty &{\text{for }}\alpha \in (0,2]\\{\frac {x_{\mathrm {m} }^{2}\alpha }{(\alpha -1)^{2}(\alpha -2)}}&{\text{for }}\alpha >2\end{cases}}}$
Skewness	${\displaystyle {\frac {2(1+\alpha )}{\alpha -3}}\,{\sqrt {\frac {\alpha -2}{\alpha }}}{\text{ for }}\alpha >3}$
Ex. kurtosis	${\displaystyle {\frac {6(\alpha ^{3}+\alpha ^{2}-6\alpha -2)}{\alpha (\alpha -3)(\alpha -4)}}{\text{ for }}\alpha >4}$
Entropy	${\displaystyle \log \left(\left({\frac {x_{\mathrm {m} }}{\alpha }}\right)\,e^{1+{\tfrac {1}{\alpha }}}\right)}$
MGF	${\displaystyle \alpha (-x_{\mathrm {m} }t)^{\alpha }\Gamma (-\alpha ,-x_{\mathrm {m} }t){\text{ for }}t<0}$
CF	${\displaystyle \alpha (-ix_{\mathrm {m} }t)^{\alpha }\Gamma (-\alpha ,-ix_{\mathrm {m} }t)}$
Fisher information	${\displaystyle {\begin{pmatrix}{\frac {\alpha }{x_{m}^{2}}}&-{\frac {1}{x_{m}}}\\-{\frac {1}{x_{m}}}&{\frac {1}{\alpha ^{2}}}\end{pmatrix}}}$

Bounded Pareto

Parameters	${\displaystyle L>0}$ location (real) ${\displaystyle H>L}$ location (real) ${\displaystyle \alpha >0}$ shape (real)
Support	${\displaystyle L\leqslant x\leqslant H}$
PDF	${\displaystyle {\frac {\alpha L^{\alpha }x^{-\alpha -1}}{1-\left({\frac {L}{H}}\right)^{\alpha }}}}$
CDF	${\displaystyle {\frac {1-L^{\alpha }x^{-\alpha }}{1-\left({\frac {L}{H}}\right)^{\alpha }}}}$
Mean	${\displaystyle {\frac {L^{\alpha }}{1-\left({\frac {L}{H}}\right)^{\alpha }}}\cdot \left({\frac {\alpha }{\alpha -1}}\right)\cdot \left({\frac {1}{L^{\alpha -1}}}-{\frac {1}{H^{\alpha -1}}}\right),\alpha \neq 1}$
Median	${\displaystyle L\left(1-{\frac {1}{2}}\left(1-\left({\frac {L}{H}}\right)^{\alpha }\right)\right)^{-{\frac {1}{\alpha }}}}$
Variance	${\displaystyle {\frac {L^{\alpha }}{1-\left({\frac {L}{H}}\right)^{\alpha }}}\cdot \left({\frac {\alpha }{\alpha -2}}\right)\cdot \left({\frac {1}{L^{\alpha -2}}}-{\frac {1}{H^{\alpha -2}}}\right),\alpha \neq 2}$ (this is the second moment, NOT the variance)
Skewness	${\displaystyle {\frac {L^{\alpha }}{1-\left({\frac {L}{H}}\right)^{\alpha }}}\cdot {\frac {\alpha *(L^{k-\alpha }-H^{k-\alpha })}{(\alpha -k)}},\alpha \neq j}$ (this is a formula for the kth moment, NOT the skewness)

The Pareto distribution, named after the Italian civil engineer, economist, and sociologist 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, 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

It follows (by differentiation) that the probability density function 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.

The conditional probability distribution of a Pareto-distributed random variable, given the event that it is greater than or equal to a particular number ${\displaystyle x_{1}}$ exceeding ${\displaystyle x_{\text{m}}}$, is a Pareto distribution with the same Pareto index ${\displaystyle \alpha }$ but with minimum ${\displaystyle x_{1}}$ instead of ${\displaystyle x_{\text{m}}}$.