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. The hyperbolic secant function is equivalent to the reciprocal hyperbolic cosine, and thus this distribution is also called the inverse-cosh distribution.

hyperbolic secant

Probability density function
Cumulative distribution function
Parameters	none
Support	${\displaystyle x\in (-\infty$ ;+\infty )\!}
PDF	${\displaystyle {\frac {1}{2}}\;\operatorname {sech} \!\left({\frac {\pi }{2}}\,x\right)\!}$
CDF	${\displaystyle {\frac {2}{\pi }}\arctan \!\left[\exp \!\left({\frac {\pi }{2}}\,x\right)\right]\!}$
Mean	${\displaystyle 0}$
Median	${\displaystyle 0}$
Mode	${\displaystyle 0}$
Variance	${\displaystyle 1}$
Skewness	${\displaystyle 0}$
Ex. kurtosis	${\displaystyle 2}$
Entropy	${\displaystyle \ln 4}$
MGF	${\displaystyle \sec(t)\!}$ for ${\displaystyle |t|<{\frac {\pi }{2}}\!}$
CF	${\displaystyle \operatorname {sech} (t)\!}$ for ${\displaystyle |t|<{\frac {\pi }{2}}\!}$

Generalisation of the distribution gives rise to the Meixner distribution, also known as the Natural Exponential Family - Generalised Hyperbolic Secant or NEF-GHS distribution.