Pareto Distribution - Definition

Definition

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

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

where xm 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 xm 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.

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