Power law vs exponential

#statistics

• Exponential: (constant)^x
• Power law: x^(constant)

Power law distributions are fractal. Twice as wealthy is half as likely, or whatever the constant you use.

Exponentials decay faster than power laws. In an exponential based on 0.5^x, every increase of x by one point halves the probability, whereas in a power law based on x^0.5, you need greater increases of x to halve the probability. Pareto distributions in particular have a property called . . .

The Pareto distribution is related to the exponential distribution as follows. If X is Pareto-distributed with minimum x_m and index α, then

$${\displaystyle Y=\log \left({\frac {X}{x_{\mathrm {m} }}}\right)}$$

is exponentially distributed with rate parameter α. Equivalently, if Y is exponentially distributed with rate α, then

$$x_\mathrm{m} e^Y$$

is Pareto-distributed with minimum x_m and index α.

This can be shown using the standard change-of-variable techniques:

$$\begin{align*} \Pr(Y<y)&=\Pr \left(\log \left({\frac {X}{x_{\mathrm {m} }}}\right)<y\right)\\&=\Pr(X<x_{\mathrm {m} }e^{y})=1-\left({\frac {x_{\mathrm {m} }}{x_{\mathrm {m} }e^{y}}}\right)^{\alpha }=1-e^{-\alpha y}. \end{align*}$$

The last expression is the cumulative distribution function of an exponential distribution with rate α.