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| {{Probability distribution |
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| name =hyperbolic secant|
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| type =density|
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| pdf_image =[[Image:Sech_distribution_PDF.png|325px|Plot of the hyperbolic secant PDF]]|
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| cdf_image =[[Image:Sech_distribution_CDF.png|325px|Plot of the hyperbolic secant CDF]]|
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| parameters =''none''|
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| support =<math>x \in (-\infty; +\infty)\!</math>|
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| pdf =<math>\frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\!</math>|
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| cdf =<math>\frac{2}{\pi} \arctan\!\left[\exp\!\left(\frac{\pi}{2}\,x\right)\right]\!</math>|
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| mean =<math>0</math>|
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| median =<math>0</math>|
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| mode =<math>0</math>|
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| variance =<math>1</math>|
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| skewness =<math>0</math>|
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| kurtosis =<math>2</math>|
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| entropy =4/''π'' ''[[Catalan's constant|K]]'' <math>\;\approx 1.16624</math>|
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| mgf =<math>\sec(t)\!</math> for <math>|t|<\frac{\pi}2\!</math>|
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| char =<math>\operatorname{sech}(t)\!</math> for <math>|t|<\frac{\pi}2\!</math>|
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| }}
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| In [[probability theory]] and [[statistics]], the '''hyperbolic secant distribution''' is a continuous [[probability distribution]] whose [[probability density function]] and [[Characteristic function (probability theory)|characteristic function]] are proportional to the [[hyperbolic function|hyperbolic secant function]]. The hyperbolic secant function is equivalent to the inverse [[Hyperbolic function|hyperbolic cosine]], and thus this distribution is also called the '''inverse-cosh distribution'''.
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| == Explanation ==
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| A [[random variable]] follows a hyperbolic secant distribution if its probability density function (pdf) can be related to the following standard form of density function by a location and shift transformation:
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| :<math>f(x) = \frac12 \; \operatorname{sech}\!\left(\frac{\pi}{2}\,x\right)\! ,</math>
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| where "sech" denotes the hyperbolic secant function.
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| The [[cumulative distribution function]] (cdf) of the standard distribution is
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| :<math>F(x) = \frac12 + \frac{1}{\pi} \arctan\!\left[\operatorname{sinh}\!\left(\frac{\pi}{2}\,x\right)\right] | |
| \! ,</math>
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| :<math> = \frac{2}{\pi} \arctan\!\left[\exp\left(\frac{\pi}{2}\,x\right)\right] \! .</math>
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| where "arctan" is the [[trigonometric function#Inverse functions|inverse (circular) tangent function]].
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| The inverse cdf (or quantile function) is
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| :<math>F^{-1}(p) = -\frac{2}{\pi}\, \operatorname{arsinh}\!\left[\cot(\pi\,p)\right] \! ,</math> | |
| :<math> = \frac{2}{\pi}\, \ln\!\left[\tan\left(\frac{\pi}{2}\,p\right)\right] \! .</math>
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| where "arsinh" is the [[inverse hyperbolic function|inverse hyperbolic sine function]] and "cot" is the [[trigonometric function|(circular) cotangent function]].
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| The hyperbolic secant distribution shares many properties with the standard [[normal distribution]]: it is symmetric with unit [[variance]] and zero [[expected value|mean]], [[median]] and [[mode (statistics)|mode]], and its pdf is proportional to its characteristic function. However, the hyperbolic secant distribution is [[kurtosis#Terminology and examples|leptokurtic]]; that is, it has a more acute peak near its mean, and heavier tails, compared with the standard normal distribution.
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| Johnson et al. (1995, p147) place this distribution in the context of a class of generalised forms of the [[logistic distribution]], but use a different parameterisation of the standard distribution compared to that here.
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| ==References==
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| * {{cite journal|first=W. D. |last=Baten |year=1934 |title=The probability law for the sum of ''n'' independent variables, each subject to the law <math>(2h)^{-1} \operatorname{sech}(\pi x/2h)</math> |journal=[[Bulletin of the American Mathematical Society]] |volume=40 |issue=4 |pages=284–290 |doi=10.1090/S0002-9904-1934-05852-X }}
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| * J. Talacko, 1956, "Perks' distributions and their role in the theory of Wiener's stochastic variables", ''[[Trabajos de Estadistica]]'' 7:159–174.
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| * Luc Devroye, 1986, [http://cgm.cs.mcgill.ca/~luc/rnbookindex.html ''Non-Uniform Random Variate Generation''], Springer-Verlag, New York. Section IX.7.2.
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| * {{Cite journal
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| | author = [[G.K. Smyth]]
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| | title = A note on modelling cross correlations: Hyperbolic secant regression
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| | journal = [[Biometrika]]
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| | volume = 81
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| | issue = 2
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| | pages = 396–402
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| | year = 1994
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| | url = http://www.statsci.org/smyth/pubs/sech.pdf
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| | doi = 10.1093/biomet/81.2.396
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| }}
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| * Norman L. Johnson, Samuel Kotz and N. Balakrishnan, 1995, ''Continuous Univariate Distributions'', volume 2, ISBN 0-471-58494-0.
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| {{ProbDistributions|continuous-infinite}}
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| [[Category:Continuous distributions]]
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| [[Category:Probability distributions]]
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