Half-logistic distribution

Probability density function
Cumulative distribution function
Support	${\displaystyle k\in [0;\infty )\!}$
PDF	${\displaystyle {\frac {2e^{-k)){(1+e^{-k})^{2))}\!}$
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,

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

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,

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