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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

$X=\left|Y\right|$

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,

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

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,

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

References

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