Half-logistic distribution

Half-logistic distribution

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

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

${\displaystyle X = |Y| \!}$

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,

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

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,

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

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" (PDF). InterStat (2).