Half-logistic distribution

In probability theory and statistics, the half-logistic distribution is a continuous probability distribution—the distribution of the absolute value of a random variable following the logistic distribution. That is, for

Half-logistic distribution

Probability density function
Cumulative distribution function
Support	$k\in [0;\infty )\!$
PDF	${\frac {2e^{-k}}{(1+e^{-k})^{2}}}\!$
CDF	${\frac {1-e^{-k}}{1+e^{-k}}}\!$
Mean	$\ln(4)=1.386\ldots$
Median	$\ln(3)=1.0986\ldots$
Mode	0
Variance	$\pi ^{2}/3-(\ln(4))^{2}=1.368\ldots$

$X=|Y|\!$

where Y is a logistic random variable, X is a half-logistic random variable.

Specification

Cumulative distribution function

The cumulative distribution function (cdf) of the half-logistic distribution is intimately related to the cdf of the logistic distribution. Formally, if F(k) is the cdf for the logistic distribution, then G(k) = 2F(k) − 1 is the cdf of a half-logistic distribution. Specifically,

$G(k)={\frac {1-e^{-k}}{1+e^{-k}}}{\text{ for }}k\geq 0.\!$

Probability density function

Similarly, the probability density function (pdf) of the half-logistic distribution is g(k) = 2f(k) if f(k) is the pdf of the logistic distribution. Explicitly,

$g(k)={\frac {2e^{-k}}{(1+e^{-k})^{2}}}{\text{ for }}k\geq 0.\!$

References

• Johnson, N. L.; Kotz, S.; Balakrishnan, N. (1994). "23.11". Continuous univariate distributions. Vol. 2 (2nd ed.). New York: Wiley. p. 150.
• George, Olusegun; Meenakshi Devidas (1992). "Some Related Distributions". In N. Balakrishnan (ed.). Handbook of the Logistic Distribution. New York: Marcel Dekker, Inc. pp. 232–234. ISBN 0-8247-8587-8.
• Olapade, A.K. (2003), "On characterizations of the half-logistic distribution" (PDF), InterStat, 2003 (February): 2, ISSN 1941-689X