# Half-logistic distribution

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

${\displaystyle X=|Y|\!}$
Support Probability density function Cumulative distribution function ${\displaystyle k\in [0;\infty )\!}$ ${\displaystyle {\frac {2e^{-k}}{(1+e^{-k})^{2}}}\!}$ ${\displaystyle {\frac {1-e^{-k}}{1+e^{-k}}}\!}$ ${\displaystyle \log _{e}(4)=1.386\ldots }$ ${\displaystyle \log _{e}(3)=1.0986\ldots }$ 0 ${\displaystyle \pi ^{2}/3-(\log _{e}(4))^{2}=1.368\ldots }$

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,

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

### 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,

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

## References

• Johnson, N. L.; Kotz, S.; Balakrishnan, N. (1994). "23.11". Continuous univariate distributions. 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