# Hyperbolic secant distribution

Parameters Probability density functionPlot of the hyperbolic secant PDF Cumulative distribution functionPlot of the hyperbolic secant CDF none ${\displaystyle x\in (-\infty ;+\infty )\!}$ ${\displaystyle {\frac {1}{2}}\;\operatorname {sech} \!\left({\frac {\pi }{2}}\,x\right)\!}$ ${\displaystyle {\frac {2}{\pi }}\arctan \!\left[\exp \!\left({\frac {\pi }{2}}\,x\right)\right]\!}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 0}$ ${\displaystyle 2}$ 4/π K ${\displaystyle \;\approx 1.16624}$ ${\displaystyle \sec(t)\!}$ for ${\displaystyle |t|<{\frac {\pi }{2}}\!}$ ${\displaystyle \operatorname {sech} (t)\!}$ for ${\displaystyle |t|<{\frac {\pi }{2}}\!}$

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

${\displaystyle f(x)={\frac {1}{2}}\;\operatorname {sech} \!\left({\frac {\pi }{2}}\,x\right)\!}$

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

${\displaystyle F(x)={\frac {1}{2}}+{\frac {1}{\pi }}\arctan \!\left[\operatorname {sech} \!\left({\frac {\pi }{2}}\,x\right)\right]\!}$
${\displaystyle ={\frac {2}{\pi }}\arctan \!\left[\exp \left({\frac {\pi }{2}}\,x\right)\right]\!}$

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

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

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 ${\displaystyle (2h)^{-1}\operatorname {sech} (\pi x/2h)}$", 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, Non-Uniform Random Variate Generation, Springer-Verlag, New York. Section IX.7.2.
• G.K. Smyth (1994). "A note on modelling cross correlations: Hyperbolic secant regression" (PDF). Biometrika. 81: 396–402. 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.