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Log-Series Distribution

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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 ln(1-theta) about theta=0. It has probability and density functions given by

P(n)=-(theta^n)/(nln(1-theta))
(1)
D(n)=1+(B(t;n+1,0))/(ln(1-t)),
(2)

where B(z;a,b) is the incomplete beta function.

The log-series distribution is implemented as LogSeriesDistribution[theta].

It is properly normalized since

-sum_(n=1)^infty(theta^n)/(nln(1-theta))=1.
(3)

The nth raw moment is given by

mu_n^'=-(Li_(1-n)(theta))/(ln(1-theta)),
(4)

where Li_n(z) is a polylogarithm.

The mean, variance, skewness, and kurtosis excess

mu=theta/((theta-1)ln(1-theta))
(5)
sigma^2=-(theta[theta+ln(1-theta)])/((theta-1)^2[ln(1-theta)]^2)
(6)
gamma_1=(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)
(7)
gamma_2=(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).
(8)

See also

Log Normal Distribution, Logarithmic Distribution

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Cite this as:

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

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