Half-logistic distribution

Probability distribution
name =Half-logistic distribution
type =density
pdf_

cdf_

parameters =
support =$k\; in\; [0;infty)!$
pdf =$frac\{2\; e^\{-k${(1+e^{-k})^2}!
cdf =$frac\{1-e^\{-k\{1+e^\{-k!$
mean =$log\_e(4)=1.386ldots$
median =$log\_e(3)=1.0986ldots$
mode =0
variance =$pi^2/3-(log\_e(4))^2=1.368ldots$
skewness =
kurtosis =
entropy =
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") = 2"F"("k") − 1 is the cdf of a half-logistic distribution. Specifically,

:$G(k)\; =\; frac\{1-e^\{-k\{1+e^\{-k\; mbox\{\; for\; \}\; kgeq\; 0.\; !$

Probability density function

Similarly, 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. Explicitly,

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

References

*cite book | last = George | first = Olusengun | coauthors = Meenakshi Devidas | editor = N. Balakrishnan | title = Handbook of the Logistic Distribution | year = 1992 | publisher = Marcel Dekker, Inc. | location = New York | pages = 232-234 | chapter = Some Related Distributions | id = ISBN 0-8247-8587-8

*cite journal | first = A.K. | last = Olapade | year = 2003 | month = February | title = "On Characterizations of the Half-Logistic Distribution" | journal = InterStat, | issue = 2 | url = http://interstat.statjournals.net/YEAR/2003/articles/0302002.pdf