Pareto Distribution

Basic Concepts

The pdf of the Pareto (type I) distribution with scale parameter m > 0 and scale parameter α > 0 is

Pareto pdf

for xm and f(x) = 0 otherwise. The corresponding cumulative distribution is

Pareto cdf

for xm and F(x) = 0 otherwise.

From the formula for the cdf, it is easy to see that the inverse function is

Pareto inverse function

Properties

Key statistical properties of the Pareto distribution are shown in Figure 1.

Pareto distribution properties

Figure 1 – Statistical properties of the Pareto distribution

For a graph of the Pareto distribution at m = 1 and α = 1, 2, 3, see Figure 2.

Chart of Pareto distribution

Figure 2 – Pareto distribution

Worksheet Functions

Real Statistics Functions: The Real Statistics Resource Pack provides the following functions for the Pareto distribution.

PARETO_DIST(x, α, m, cum) = pdf of the Pareto distribution f(x) when cum = FALSE and the corresponding cumulative distribution function F(x) when cum = TRUE.

PARETO_INV(p, α, m) = inverse of the Pareto distribution at p.

Distribution Fitting

Given a collection of data that may fit the Pareto distribution, we explore two ways to estimate the parameters that best fit the data. See the following for details: Method of Moments and MLE Fitting.

Pareto Principle

In the case where the shape parameter is α = log45 = 1.160964, we get the famous Pareto principle, aka the 80-20 rule, which states that 80% of the outcomes are due to 20% of the causes. E.g. 20% of the workers do 80% of the work. 80% of the wealth is owned by 20% of the people.

Generalized Pareto Distribution

Click here for information about a generalization of the Pareto distribution.

Examples Workbook

Click here to download the Excel workbook with the examples described on this webpage.

References

Wikipedia (2021) Pareto distribution
https://en.wikipedia.org/wiki/Pareto_distribution

Wikipedia (2021) Pareto principle
https://en.wikipedia.org/wiki/Pareto_principle

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