Half-logistic distribution

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.

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,

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