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probability theory > probability distribution > logarithmic distribution

mathematical statistics > probability distribution > logarithmic distribution

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logarithmic distribution

In probability and statistics, the logarithmic distribution (also known as the logarithmic series distribution or the log-series distribution) is a discrete probability distribution derived from the Maclaurin series expansion
${\\displaystyle -\\ln(1-p)=p+{\ rac {p^{2}}{2}}+{\ rac {p^{3}}{3}}+\\cdots .}$
From this we obtain the identity
${\\displaystyle \\sum _{k=1}^{\\infty }{\ rac {-1}{\\ln(1-p)}}\\;{\ rac {p^{k}}{k}}=1.}$
This leads directly to the probability mass function of a Log(p)-distributed random variable:
${\\displaystyle f(k)={\ rac {-1}{\\ln(1-p)}}\\;{\ rac {p^{k}}{k}}}$
for k ≥ 1, and where 0 < p < 1. Because of the identity above, the distribution is properly normalized. The cumulative distribution function is
${\\displaystyle F(k)=1+{\ rac {\\mathrm {B} (p;k+1,0)}{\\ln(1-p)}}}$
where B is the incomplete beta function.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Logarithmic_distribution)

probability distribution

logarithmic series distribution
log-series distribution

loi logarithmique

francés

URI

http://data.loterre.fr/ark:/67375/PSR-NLD0HQRC-8

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