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.

A random variable follows a hyperbolic secant distribution if its probability density function can be related to the following standard form of density function by a location and shift transformation: where "sech" denotes the hyperbolic secant function.

The cumulative distribution function (cdf) of the standard distribution is a scaled and shifted version of the Gudermannian function, where "arctan" is the inverse (circular) tangent function.

Ding (2014)[2] shows three occurrences of the Hyperbolic secant distribution in statistical modeling and inference.

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 probability density function is proportional to its characteristic function.

However, the hyperbolic secant distribution is leptokurtic; that is, it has a more acute peak near its mean, and heavier tails, compared with the standard normal distribution.

independent and identically distributed hyperbolic secant random variables: then in the limit

, in accordance with the central limit theorem.

This allows a convenient family of distributions to be defined with properties intermediate between the hyperbolic secant and the normal distribution, controlled by the shape parameter

, which can be extended to non-integer values via the characteristic function Moments can be readily calculated from the characteristic function.

The distribution (and its generalisations) can also trivially be shifted and scaled in the usual way to give a corresponding location-scale family: A skewed form of the distribution can be obtained by multiplying by the exponential

and normalising, to give the distribution where the parameter value

The Champernowne distribution has an additional parameter to shape the core or wings.

Allowing all four of the adjustments above gives distribution with four parameters, controlling shape, skew, location, and scale respectively, called either the Meixner distribution[3] after Josef Meixner who first investigated the family, or the NEF-GHS distribution (Natural exponential family - Generalised Hyperbolic Secant distribution).

In financial mathematics the Meixner distribution has been used to model non-Gaussian movement of stock-prices, with applications including the pricing of options.

Losev (1989) has studied independently the asymmetric (skewed) curve

a measure of right skew, in case the parameters are both positive.