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Half-logistic distribution

Concept in statistics

name =Half-logistic distribution| type =density| pdf_image =[[Image:Half-logistic distribution pdf.svg|325px|Probability density plots of half-logistic distribution]]| cdf_image =[[Image:Half-logistic distribution cdf.svg|325px|Cumulative distribution plots of half-logistic distribution]]| parameters =| support =k \in 0;\infty)!| pdf =\frac{2 e^{-k}}{(1+e^{-k})^2}!| cdf =\frac{1-e^{-k}}{1+e^{-k}}!| mean =\ln(4)=1.386\ldots| median =\ln(3)=1.0986\ldots| mode =0| variance =\pi^2/3-(\ln(4))^2=1.368\ldots| skewness =| kurtosis =| entropy =\log\left(\frac{e^{2}}{2}\right)| mgf =| char =|

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

References

Info: Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

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