  It is argued that there is a need for fat-tailed distributions that become
thin in the extreme tail. A 3-parameter distribution is introduced that
visually resembles the t-distribution and interpolates between the normal
distribution and the Cauchy distribution. It is fat-tailed, but has all moments
finite, and the moment-generating function exists. It would be useful as an
alternative to the t-distribution for a sensitivity analysis to check the
robustness of results or for computations where finite moments are needed, such
as in option-pricing. It can be motivated probabilistically in at least two
ways, either as the random thinning of a long-tailed distribution, or as random
variation of the variance of a normal distribution. Its properties are
described, algorithms for random-number generation are provided, and examples
of its use in data-fitting given. Some related distributions are also
discussed, including asymmetric and multivariate distributions.
