Half-logistic distribution

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Half-logistic distribution
Probability density function
Probability density plots of half-logistic distribution
Cumulative distribution function
Cumulative distribution plots of half-logistic distribution
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.


Cumulative distribution function[edit]

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[edit]

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. \!


  • 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.