Logarithmic Distribution
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8-4-2021
2026
Logarithmic Distribution
The logarithmic distribution is a continuous distribution for a variate ![X in [a,b]](https://mathworld.wolfram.com/images/equations/LogarithmicDistribution/Inline1.gif)
with probability function
 |
(1)
|
and distribution function
 |
(2)
|
It therefore applies to a variable distributed as 
, 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
 |
(3)
|
The mean is therefore
![mu=(a^2(1-2lna)-b^2(1-2lnb))/(4[a(1-lna)-b(1-lnb)]).](https://mathworld.wolfram.com/images/equations/LogarithmicDistribution/NumberedEquation4.gif) |
(4)
|
The variance, skewness, and kurtosis excess are slightly complicated expressions.
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