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Log Uniform Distribution

A log uniform distribution is a continuous distribution of a positive random variable X for which lnX has a uniform distribution on [lna,lnb], where 0<a<b. It is also called the reciprocal distribution (SciPy). Here, ln denotes the natural logarithm.

Its probability density function is

f(x)=1/(xln(b/a))
(1)

for a<=x<=b, and is zero outside this interval. Its cumulative distribution function is

F(x)={0 for x<a; ln(x/a)/ln(b/a) for a<=x<=b; 1 for x>b.
(2)

The raw moments are

mu_n^'=(b^n-a^n)/(nln(b/a))
(3)

for positive integers n. In particular, the mean is (b-a)/ln(b/a).

For a=1 and b=10, the leading decimal digits follow Benford's law. This density is distinct from the density proportional to lnx called the logarithmic distribution in this work.

See also

Benford's Law, Logarithmic Distribution, Uniform Distribution

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References

SciPy. "scipy.stats.loguniform." SciPy Manual. https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.loguniform.html.

Cite this as:

Weisstein, Eric W. "Log Uniform Distribution." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LogUniformDistribution.html

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