What is the Pareto distribution?

The Pareto distribution is a continuous probability distribution used to model power law phenomena.

The Pareto distribution is named after the Italian economist Vilfredo Pareto, who observed that 80% of the land in Italy was owned by 20% of the population. This observation led to the development of the Pareto principle, which states that 80% of the effects come from 20% of the causes.

The Pareto distribution is a power law distribution, which means that it has a long tail that extends to infinity. The probability density function (PDF) of the Pareto distribution is given by:

f(x) = (αxₘᵢⁿ) / x^(α+1)

where x is the random variable, xₘᵢⁿ is the minimum value of x, and α is the shape parameter. The mean and variance of the Pareto distribution are given by:

μ = (αxₘᵢⁿ) / (α-1)

σ² = (αxₘᵢⁿ)² / ((α-1)²(α-2))

The Pareto distribution is commonly used to model income and wealth distributions, as well as the distribution of city sizes, earthquake magnitudes, and many other phenomena that exhibit power law behavior. It is important to note that the Pareto distribution is only valid for values of x greater than or equal to xₘᵢⁿ.