Pareto Distribution (Continuous)¶
A specific parametric form of a power-law distribution.
Definition & Parameters
| Symbol | Meaning |
|---|---|
| \(x_m\) | Minimum value (scale parameter) — support \(x \ge x_m\) |
| \(\alpha\) | Shape parameter (tail index) — controls tail heaviness |
\(f(x) = \frac{\alpha x_m^\alpha}{x^{\alpha+1}}, \qquad x \ge x_m\)
Properties
| Property | Formula / Note |
|---|---|
| Mean (if \(\alpha>1\)) | \(\dfrac{\alpha x_m}{\alpha - 1}\) |
| Variance (if \(\alpha>2\)) | \(\dfrac{\alpha x_m^2}{(\alpha - 1)^2(\alpha - 2)}\) |
| Median | \(x_m 2^{1/\alpha}\) |
| Tail probability | \(P(X > x) = \left(\dfrac{x_m}{x}\right)^\alpha\) |
Use cases
| Domain | Example |
|---|---|
| Economics | Income / wealth distributions (Pareto principle) |
| Reliability | Large claim sizes in insurance |
| Computer networks | File size distributions, degree distributions in networks |