Half-logistic distribution
# Half-logistic distribution

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,

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

### 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,

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