Half-logistic distribution

{{Probability distribution|

name =Half-logistic distribution|

type =density|

pdf_image =Image:Half-logistic distribution pdf.svg|

cdf_image =Image:Half-logistic distribution cdf.svg|

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 =|

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

{{cite book|last1=Johnson|first1=N. L.|last2=Kotz|first2=S.|last3=Balakrishnan|first3=N.|title=Continuous univariate distributions|edition=2nd|volume=2|publisher=Wiley|place=New York|year=1994|chapter=23.11|page=150}}
{{cite book | last = George | first = Olusegun |author2=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 | isbn = 0-8247-8587-8}}
{{citation | last = Olapade | first = A.K. | year=2003 | title = On characterizations of the half-logistic distribution | journal = InterStat | volume=2003 | issue = February | page=2 | url = http://interstat.statjournals.net/YEAR/2003/articles/0302002.pdf | issn=1941-689X }}

Category:Continuous distributions