The log-series distribution, also sometimes called the logarithmic distribution (although this work reserves that term for a distinct distribution), is the distribution of the terms in the series expansion of 
about 
. It has probability and density functions given by
 |  |  | (1) |
 |  |  | (2) |
where 
is the incomplete beta function.
The log-series distribution is implemented as LogSeriesDistribution[theta].
It is properly normalized since
 | (3) |
The 
th raw moment is given by
 | (4) |
where 
is a polylogarithm.
The mean, variance, skewness, and kurtosis excess
 |  |  | (5) |
 |  | ![-(theta[theta+ln(1-theta)])/((theta-1)^2[ln(1-theta)]^2)](http://mathworld.wolfram.com/images/equations/Log-SeriesDistribution/Inline17.svg) | (6) |
 |  | ![(2theta^2+3thetaln(1-theta)+(1+theta)ln^2(1-theta))/(ln(1-theta)[theta+ln(1-theta)]sqrt(-theta[theta+ln(1-theta)]))ln(1-theta)](http://mathworld.wolfram.com/images/equations/Log-SeriesDistribution/Inline20.svg) | (7) |
 |  | ![(6theta^3+12theta^2ln(1-theta)+theta(7+4theta)ln^2(1-theta)+(1+4theta+theta^2)ln^3(1-theta))/(theta[theta+ln(1-theta)]^2).](http://mathworld.wolfram.com/images/equations/Log-SeriesDistribution/Inline23.svg) | (8) |
