  In this work some advances in the theory of curvature of two-dimensional
probability manifolds corresponding to families of distributions are proposed.
It is proved that location-scale distributions are hyperbolic in the
Information Geometry sense even when the generatrix is non-even or non-smooth.
A novel formula is obtained for the computation of curvature in the case of
exponential families: this formula implies some new flatness criteria in
dimension 2. Finally, it is observed that many two parameter distributions,
widely used in applications, are locally hyperbolic, which highlights the role
of hyperbolic geometry in the study of commonly employed probability manifolds.
These results have benefited from the use of explainable computational tools,
which can substantially boost scientific productivity.
