The Pareto Distribution
The Pareto distribution, specifically the Pareto Type I distribution, is a continuous probability distribution used to model various types of data. Here, we focus on the distribution with the parameter \(x_m\).
Probability Density Function (PDF)
The probability density function of a Pareto random variable with parameters \(\alpha\) (shape parameter) and \(x_m\) (scale parameter) is: \[ f_X(x) = \begin{cases} \alpha \dfrac{x_m^\alpha}{x^{\alpha+1}} & \text{for } x \ge x_m, \\ 0 & \text{for } x < x_m. \end{cases} \]
Cumulative Distribution Function (CDF)
The cumulative distribution function is: \[ F_X(x) = \begin{cases} 1 - \left(\dfrac{x_m}{x}\right)^\alpha & \text{for } x \ge x_m, \\ 0 & \text{for } x < x_m. \end{cases} \] When plotted on linear axes, the CDF assumes a J-shaped curve, which approaches the axes asymptotically. In a log-log plot, it is represented by a straight line.
Quantile Function
The quantile function of a Pareto random variable is: \[ F^{-1}(p) = x_m (1 - p)^{-1/\alpha}, \quad 0 \le p < 1 \]
Moments and Characteristics
Expected Value
The expected value of a Pareto random variable is: \[ E(X) = \begin{cases} \infty & \text{if } \alpha \le 1, \\ \dfrac{\alpha x_m}{\alpha - 1} & \text{if } \alpha > 1. \end{cases} \]
Variance
The variance of a Pareto random variable is: \[ \mathrm{Var}(X) = \begin{cases} \infty & \text{if } \alpha \le 2, \\ \dfrac{(\alpha x_m)^2}{(\alpha - 1)^2 (\alpha - 2)} & \text{if } \alpha > 2. \end{cases} \]
This distribution is particularly useful in modeling data where the “80-20 rule” (Pareto principle) applies, such as in economics, finance, and other fields where large values are observed infrequently but have significant impact.