Pareto Distribution#

Univariate, Continuous, Asymmetric, Non-Negative, Heavy-tailed

The Pareto distribution is a continuous probability distribution, known for following a power-law and having a heavy right tail. It is defined by two parameters: the scale parameter \(m\) and the shape parameter \(\alpha\).

It was originally used to describe the distribution of wealth in society where a small proportion of the population holds a large proportion of the wealth (the “80-20 rule”). It has since been used in various fields to describe a wide range of phenomena where events get rarer at greater magnitudes.

Key properties and parameters#

Support

\(x \in [m, \infty)\)

Mean

\(\frac{\alpha m}{\alpha - 1}\) for \(\alpha > 1\)

Variance

\(\frac{m^2 \alpha}{(\alpha - 1)^2 (\alpha - 2)}\) for \(\alpha > 2\)

Parameters:

  • \(m\) : (float) Scale parameter, \(m > 0\).

  • \(\alpha\) : (float) Shape parameter, \(\alpha > 1\).

Probability Density Function (PDF)#

\[ f(x|\alpha, m) = \frac{\alpha m^\alpha}{x^{\alpha + 1}} \]
Pareto Distribution PDF

Cumulative Distribution Function (CDF)#

\[ F(x|\alpha, m) = 1 - \left(\frac{m}{x}\right)^\alpha \]
Pareto Distribution CDF

See also

Related Distributions:

  • Exponential Distribution - If X is Pareto distributed, with scale parameter \(m\), and shape parameter \(\alpha\), then \(Y = log(X/m)\) is exponentially distributed with rate parameter \(\lambda = \alpha\).

  • Log-Normal Distribution - Also used for modeling positive, skewed data with long tails. The log-normal distribution allows for more flexibility in shaping the tail behavior compared to the Pareto distribution.

References#