Half-logistic distribution explained

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) = (1- e ^{-k})/(1+ e ^{-k}) 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) = (2 e ^{-k})/(( 1+ e ^{-k} ) ^{2}) for k\geq 0 .

References

• Book: Johnson. N. L.. Kotz. S.. Balakrishnan. N.. Continuous univariate distributions. 2nd. 2. Wiley. New York. 1994. 23.11. 150.
• Book: George, Olusegun . Meenakshi Devidas . N. Balakrishnan . Handbook of the Logistic Distribution . 1992 . Marcel Dekker, Inc. . New York . 232–234 . Some Related Distributions . 0-8247-8587-8.