# Half-logistic distribution

In probability theory and statistics, the half-logistic distribution is a continuous probability distributionthe distribution of the absolute value of a random variable following the logistic distribution. That is, for

$X=|Y|\!$ Support Probability density function Cumulative distribution function $k\in [0;\infty )\!$ ${\frac {2e^{-k}}{(1+e^{-k})^{2}}}\!$ ${\frac {1-e^{-k}}{1+e^{-k}}}\!$ $\log _{e}(4)=1.386\ldots$ $\log _{e}(3)=1.0986\ldots$ 0 $\pi ^{2}/3-(\log _{e}(4))^{2}=1.368\ldots$ 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,

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

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