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.

A random variable follows a hyperbolic secant distribution if its probability density function (pdf) is

f(x,a,b) = sech((x-a)/b)/(pib)

where “sech” denotes the hyperbolic secant 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.

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