  Let $\mathcal{N}$ be the space of Gaussian distribution functions over
$\mathbb{R}$, regarded as a 2-dimensional statistical manifold parameterized by
the mean $\mu$ and the deviation $\sigma$. In this paper we show that the
tangent bundle of $\mathcal{N}$, endowed with its natural K\"ahler structure,
is the Siegel-Jacobi space appearing in the context of Number Theory and Jacobi
forms. Geometrical aspects of the Siegel-Jacobi space are discussed in detail
(completeness, curvature, group of holomorphic isometries, space of K\"ahler
functions, relationship to the Jacobi group), and are related to the quantum
formalism in its geometrical form, i.e., based on the K\"ahler structure of the
complex projective space. This paper is a continuation of our previous work,
where we studied the quantum formalism from a geometric and
information-theoretical point of view.
