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Logarithmic Distribution


LogDistribution

The logarithmic distribution is a continuous distribution for a variate X in [a,b] with probability function

 P(x)=(lnx)/(b(lnb-1)-a(lna-1))
(1)

and distribution function

 D(x)=(a(1-lna)-x(1-lnx))/(a(1-lna)-b(1-lnb)).
(2)

It therefore applies to a variable distributed as lnx, and has appropriate normalization.

Note that the log-series distribution is sometimes also known as the logarithmic distribution, and the distribution arising in Benford's law is also "a" logarithmic distribution.

The raw moments are given by

 mu_n^'=(a^(n+1)[1-(n+1)lna]-b^(n+1)[1-(n+1)lnb])/((n+1)^2[a(1-lna)-b(1-lnb)]).
(3)

The mean is therefore

 mu=(a^2(1-2lna)-b^2(1-2lnb))/(4[a(1-lna)-b(1-lnb)]).
(4)

The variance, skewness, and kurtosis excess are slightly complicated expressions.


See also

Benford's Law, Log Normal Distribution, Log-Series Distribution

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Cite this as:

Weisstein, Eric W. "Logarithmic Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LogarithmicDistribution.html

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