# Hyperbolic secant distribution

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

$f(x)={\frac {1}{2}}\;\operatorname {sech} \!\left({\frac {\pi }{2}}\,x\right)\!$ where "sech" denotes the hyperbolic secant function. The cumulative distribution function (cdf) is

$F(x)={\frac {1}{2}}+{\frac {1}{\pi }}\arctan \!\left[\operatorname {sech} \!\left({\frac {\pi }{2}}\,x\right)\right]\!$ $={\frac {2}{\pi }}\arctan \!\left[\exp \left({\frac {\pi }{2}}\,x\right)\right]\!$ where "arctan" is the inverse (circular) tangent function. The inverse cdf (or quantile function) is

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