Half-logistic distribution

Half-logistic distribution

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

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

$X = |Y| \!$

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,

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

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,

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

References

• George, Olusengun; 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. (February 2003). "On Characterizations of the Half-Logistic Distribution". InterStat, (2). http://interstat.statjournals.net/YEAR/2003/articles/0302002.pdf.