Hyperbolic secant distribution

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


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

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\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.