The Hyperbolic Secant distribution is a bell-shaped, symmetric member of the exponential family with a mean of 0 and variance of 1. It is a continuous distribution that is tractable, which means calculation of the distribution at any point takes polynomial-time.
Hyperbolic Secant Distribution PDF
A random variable X = ln|Y1/Y2| follows a hyperbolic secant distribution (where Y are normal random variables). The density of the distribution is proportional to the hyperbolic secant function, the reciprocal of the hyperbolic cosine function defined by :
cosh(x) = 0.5 [ex + e-x]
The probability density function of the hyperbolic secant is :
Where μ ∈ ℝ and σ > 0 and
The Hyperbolic Secant distribution is not as well known as other exponential family distributions due to its isolation from many well known statistical models . It does bear some similarities to the normal distribution both in shape and symmetry. Both distributions have a density proportional to their characteristic functions although the hyperbolic secant has slightly heavier tails .
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 Stat 5601 (Geyer) Hyperbolic Secant Distribution. Retrieved December 18, 2021 from: https://www.stat.umn.edu/geyer/old02/5601/examp/hsec.html
 Rubio, F. The Hyperbolic Secant Distribution. Retrieved December 18, 2021 from: https://rpubs.com/FJRubio/HSD
 Ding, P. (2014). Three Occurrences of the Hyperbolic-Secant Distribution. Retrieved December 18, 2021 from: https://arxiv.org/abs/1401.1267
 M. J. Fischer, Generalized Hyperbolic Secant Distributions, 1
SpringerBriefs in Statistics, DOI: 10.1007/978-3-642-45138-6_1