Editing Pareto distribution (section)

==Definitions==
If ''X'' is a [[random variable]] with a Pareto (Type I) distribution,<ref name=arnold>{{cite book |first=Barry C. |last=Arnold |year=1983 |title=Pareto Distributions |publisher=International Co-operative Publishing House |isbn= 978-0-89974-012-6}}</ref> then the probability that ''X'' is greater than some number ''x'', i.e., the [[survival function]] (also called tail function), is given by

:<math>\overline{F}(x) = \Pr(X>x) = \begin{cases}
\left(\frac{x_\mathrm{m}}{x}\right)^\alpha & x\ge x_\mathrm{m}, \\
1 & x < x_\mathrm{m},
\end{cases}
</math>

where ''x''<sub>m</sub> is the (necessarily positive) minimum possible value of ''X'', and ''α'' is a positive parameter. The type I Pareto distribution is characterized by a [[scale parameter]] ''x''<sub>m</sub> and a [[shape parameter]] ''α'', which is known as the ''tail index''. If this distribution is used to model the distribution of wealth, then the parameter ''α'' is called the [[Pareto index]].

===Cumulative distribution function===
From the definition, the [[cumulative distribution function]] of a Pareto random variable with parameters ''α'' and ''x''<sub>m</sub> is

:<math>F_X(x) = \begin{cases}
1-\left(\frac{x_\mathrm{m}}{x}\right)^\alpha & x \ge x_\mathrm{m}, \\
0 & x < x_\mathrm{m}.
\end{cases}</math>

===Probability density function===
It follows (by [[Derivative|differentiation]]) that the [[probability density function]] is

:<math>f_X(x)= \begin{cases} \frac{\alpha x_\mathrm{m}^\alpha}{x^{\alpha+1}} & x \ge x_\mathrm{m}, \\ 0 & x < x_\mathrm{m}. \end{cases} </math>

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.