Power law vs exponential

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

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

is Pareto-distributed with minimum x_m and index α.

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

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

Created 2022-Jan-19 (3 years ago)