Half-logistic distribution

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Half-Logistic Distribution
Probability density function
Probability density plots of half-logistic distribution
Cumulative distribution function
Cumulative distribution plots of half-logistic distribution
Parameters
Support k \in [0;\infty)\!
Probability density function (pdf) \frac{2 e^{-k}}{(1+e^{-k})^2}\!
Cumulative distribution function (cdf) \frac{1-e^{-k}}{1+e^{-k}}\!
Mean \log_e(4)=1.386\ldots
Median \log_e(3)=1.0986\ldots
Mode 0
Variance \pi^2/3-(\log_e(4))^2=1.368\ldots
Skewness
Excess kurtosis
Entropy
Moment-generating function (mgf)
Characteristic function


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.

Contents

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) = 2 F(k) - 1 \! is the cdf of a half-logistic distribution. Specifically,

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

Similarily, the probability density function (pdf) of the half-logistic distribution is g(k) = 2 f(k) \! if f(k) \! is the pdf of the logistic distribution. Explicity,

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

  • George, Olusengun; Meenakshi Devidas (1992). "Some Related Distributions", in N. Balakrishnan: Handbook of the Logistic Distribution. New York: Marcel Dekker, Inc., 232-234. ISBN 0-8247-8587-8. 
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