The Pareto probability distribution has density

for *x* ≥ 1 where *a* > 0 is a shape parameter. The Pareto distribution and the Pareto principle (i.e. “80-20” rule) are named after the same person, the Italian economist Vilfredo Pareto.

Samples from a Pareto distribution obey Benford’s law in the limit as the parameter *a* goes to zero. That is, the smaller the parameter *a*, the more closely the distribution of the first digits of the samples come to following the distribution known as Benford’s law.

Here’s an illustration of this comparing the distribution of 1,000 random samples from a Pareto distribution with shape *a* = 1 and shape *a* = 0.2 with the counts expected under Benford’s law.

Note that this has nothing to do with base 10 per se. If we look at the leading digits as expressed in any other base, such as base 16 below, we see the same pattern.

## More posts on Benford’s law

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