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
where Y is a logistic random variable, X is a half-logistic random variable.

Specification

Cumulative distribution function

The cumulative distribution function of the half-logistic distribution is intimately related to the cdf of the logistic distribution. Formally, if F is the cdf for the logistic distribution, then G = 2F − 1 is the cdf of a half-logistic distribution. Specifically,

Probability density function

Similarly, the probability density function of the half-logistic distribution is g = 2f if f is the pdf of the logistic distribution. Explicitly,