The logarithmic distribution is a continuous distribution for a variate with probability function
(1)
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and distribution function
(2)
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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)
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The mean is therefore
(4)
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The variance, skewness, and kurtosis excess are slightly complicated expressions.