Hyperbolic secant distribution

Probability distribution
name =hyperbolic secant
type =density
pdf_

cdf_

parameters ="none"
support =$x\; in\; (-infty;\; +infty)!$
pdf =$frac12\; ;\; operatorname\{sech\}!left(frac\{pi\}\{2\},x\; ight)!$
cdf =$frac\{2\}\{pi\}\; arctan!left\; [exp!left(frac\{pi\}\{2\},x\; ight)\; ight]\; !$
mean =$0$
median =$0$
mode =$0$
variance =$1$
skewness =$0$
kurtosis =$2$
entropy =4/"π" "K" $;approx\; 1.16624$
mgf =$sec(t)!$ for
char =$operatorname\{sech\}(t)!$ for
In probability theory and statistics, the hyperbolic secant distribution is a continuous probability distribution whose probability density function and characteristic function are proportional to the hyperbolic secant function.

Explanation

A random variable follows a hyperbolic secant distribution if its probability density function (pdf) is

:$f(x)\; =\; frac12\; ;\; operatorname\{sech\}!left(frac\{pi\}\{2\},x\; ight)!$

where "sech" denotes the hyperbolic secant function.The cumulative distribution function (cdf) is

:$F(x)\; =\; frac12\; +\; frac\{1\}\{pi\}\; arctan!left\; [operatorname\{sech\}!left(frac\{pi\}\{2\},x\; ight)\; ight]\; !$:$=\; frac\{2\}\{pi\}\; arctan!left\; [expleft(frac\{pi\}\{2\},x\; ight)\; ight]\; !$

where "arctan" is the inverse (circular) tangent function.The inverse cdf (or quantile function) is

:$F^\{-1\}(p)\; =\; -frac\{2\}\{pi\},\; operatorname\{arcsinh\}!left\; [cot(pi,p)\; ight]\; !$

where "arcsinh" is the inverse hyperbolic sine function and "cot" is the (circular) cotangent function.

The hyperbolic secant distribution shares many properties with the standard normal distribution: it is symmetric with unit variance and zero mean, median and mode, and its pdf is proportional to its characteristic function. However, the hyperbolic secant distribution is leptokurtic, that is, it has a more acute peak near its mean, compared with the standard normal distribution.

References

* W. D. Baten, 1934, "The probability law for the sum of "n" independent variables, each subject to the law $(2h)^\{-1\}\; operatorname\{sech\}(pi\; x/2h)$", "Bulletin of the American Mathematical Society" 40: 284–290.
* J. Talacko, 1956, "Perks' distributions and their role in the theory of Wiener's stochastic variables", "Trabajos de Estadistica" 7:159–174.
* Luc Devroye, 1986, [http://cgm.cs.mcgill.ca/~luc/rnbookindex.html "Non-Uniform Random Variate Generation"] , Springer-Verlag, New York. Section IX.7.2.
* Cite journal
author = G.K. Smyth
title = A note on modelling cross correlations: Hyperbolic secant regression
journal = Biometrika
volume = 81
pages = 396–402
year = 1994
url = http://www.statsci.org/smyth/pubs/sech.pdf
doi = 10.1093/biomet/81.2.396
* Norman L. Johnson, Samuel Kotz and N. Balakrishnan, 1995, "Continuous Univariate Distributions", volume 2, ISBN 0-471-58494-0.