Half-logistic distribution

Half-logistic distribution

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

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

[math]\displaystyle{ X = |Y| \! }[/math]

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,

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

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,

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

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. 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", InterStat 2003 (February): 2, ISSN 1941-689X, http://interstat.statjournals.net/YEAR/2003/articles/0302002.pdf