Power-law distribution ($p_X(x)=C{x}^{-\beta }$)

A power-law distribution uses a power law for a probability mass function or a probability density on a specified range. For a continuous model with $p_X(x)=C{x}^{-\beta }$ on $x\ge x_{\min }>0$ and zero below it, the total integral is finite exactly when $\beta >1$. Normalization then requires $C=(\beta -1)x_{\min }^{\beta -1}$, giving a Pareto distribution with shape parameter $\alpha =\beta -1$. Discrete models instead normalize a sum. The support and normalizing coefficient are part of the model, and a power law for its tail probability uses a different exponent from its probability density.